Fermat's Enigma, the sublime in all its states

Note to the reader. No need to know mathematics to enjoy the tricks of the magistrate Pierre de Fermat. On the contrary.

Last update, March 19, 2023.

French version here, completed, November 11, 2023.

Sage parmi les fous
dans la cité la rumeur
et le ciel d'azur

Pierre de Fermat according to the author.

"This remarkable man, the first of us, sent me two very subtle propositions, without accompanying them with their demonstrations. And when I asked him for the demonstrations of these difficult propositions, he answered me, by letter, in these terms: "I had to work to discover them. You too work; you will thus become aware that it is in this work that the greater part of the pleasure consists." Roberval evoking Fermat around 1646, in a letter to Torricelli.
"The deep and shared conviction that Fermat did not possess a proof of his theorem comes from the long history of attempts to establish it. [...] Followers of followers, in all situations of this kind, don't know anything about what had motivated the founders [...]. They think that they know everything there is to know, all from the beginning."Jacques Roubaud, “Mathématique:” (1997)
"Anyway, this approach [by Andrew Wiles] where Fermat's theorem is only a very attractive but minor corollary, relies on recent Galois representation techniques. It remains possible that a direct elementary proof can be found." Catherine Goldstein (1995).
"As Fermat's proof is no longer necessary today, was it sufficient at the time?" Roland Franquart in 2008.

This study is based much more on pure reflection, logic, than on mathematics. There is only one calculation, a short multiplication (10 times 64). If you like big and beautiful puzzles, and if you have a level in mathematics, let's say 9^th or 8^th grade, it is more than enough. The first steps of this study were taken a long time ago, in 2006. It was initiated on fr.wikiversity in 2019 and completed on March 10, 2023. If you like big and beautiful enigmas, and if you have a level in mathematics, let's say Year 9 or 8th grade, it is more than enough. With Mr. Roland Franquart, who in 2009 solved this puzzle that dated from 1670, we have made an important study of it. The enigma had never been decrypted before R. F. became aware of it and took an interest in it, which - also - what makes it so interesting. What professional mathematician could have imagined that Fermat could have encrypted his famous observation in three and a half lines of Latin writing? Who else but a Latinist, a great lover of mathematics (R. F.), a former military man, an innovative technician working in... radars (!), could have brought it to light? The greatest attraction of this enigma is for me the masterly feat of encryption achieved by Fermat, the "Prince of amateurs". This study was written for you, and for my pleasure of course. After giving me so many happy surprises, so many joys, it is now finished. Apart from Fermat, I don't think a mathematician has ever gone so far into the deep, intimate understanding of numbers; I am convinced that never has a mathematician gone as far as Pierre Fermat in the deep and intimate understanding of the number; all the more convinced after having read the book written by Catherine Goldstein, A theorem of Fermat and its readers. I must attest here, in a fundamental ethical duty, that I am eternally grateful to Pierre Fermat. Many thanks to all my readers, your frequent visits encouraged and stimulated me, without you, this study could not have been as complete, as successful.

Claude Mariotti, March 10, 2023

Foreword

Before the creation of the French Academy of Sciences by Colbert in 1666, there was no mathematical institution in France, the custom among mathematicians was to challenge each other. As we'll see later Fermat had also tested 6 of his main correspondents about his famous conjecture on what would later be called "Fermat's numbers", for which he had invented the method allowing him in a few minutes to show that it was false. The facetious genius announced with aplomb that he was (almost) sure of its truth, but he asked them... to provide him with a proof of its validity. For 19 years, from 1640 to 1659, one by one, he tested them in this way. None of them having been able to prove (or invalidate) the conjecture, he was able to see the limits of their skills in the "science of numbers" (number theory). This was also the case for most of his other challenges, which nobody wanted (could) to take up. It has even been claimed that all these problems he proposed had absolutely no interest. In spite of the ease (for Fermat in any case) of proving the falsity of the conjecture, it was not until 1732 that the young Euler succeeded. Finally, no mathematician contemporary of Fermat wanted to follow him - even though this would have greatly contributed to the progress of science. He was saddened, bitter, very frustrate. His response was as pedagogical as his reputation: he challenged the world. According to us, his proof being so admirably hidden thanks to the Latin language, he must have been quite sure that it would take a (very?) long time before people would realize that he really "has unveiled" it (sane detexi). And the mathematicians who, according to him, had well deserved it, would have plenty to struggle with, tear their hair out, and get angry about for a long time!

Fermat's Great Theorem

"Do you want to know what is most honorable? Like to be ignored." Final words on Fermat's epitaph.

What makes the problem fascinating, although it is very difficult to solve, is the simplicity of its statement. For more than three centuries the greatest mathematicians in the world have tried in vain to prove the veracity of this theorem, that Pierre Fermat (or Pierre de Fermat) says to "really have unveiled the astonishing explanation”. If this Himalayan of mathematics could be climbed after 324 years of efforts and disappointed hopes by Andrew Wiles in 1994, it's only by modern means, a very indirect way, with a demonstration of enormous complexity, a thousand pages long in the first copy. On June 24, 1993, the day after the conference where Andrew Wiles wrote on the blackboard how he thinks he has proved the truthfulness of "Fermat's last theorem", The New York Times entitled: "At Last, Shout of 'Eureka' In Age-Old Math Mystery". The much shorter proof (although very difficult to assimilate) given by Fermat has still not been accepted by our modern mathematicians.

As soon as I learned of the existence of Pierre de Fermat's formidable enigma, so full of history, I perceived (I'm sure I'm not the only one) that this great polymath was not only very sincere and very sure of himself, but also very inventive, very secretive and, icing on the cake, very mischievous. I also had a not inconsiderable advantage, having as my "ally" in this affair, Blaise Pascal - no less - who gave an extraordinary praise to Pierre de Fermat. I smile, dear reader, thinking that like me, you will discover fascinating things (I hope you will find them interesting, at least!) about an immense 17th century genius, things that scholars all over the world will never read - or they will be extremely rare, I know only 5 of them. But I would be very surprised if they discuss it a lot among themselves, most of them not interested, too busy with their personal work. If you appreciate this study, dear reader, then perhaps you will smile, as I do, realizing that it is not necessary to be a great scientist to discover admirable things that would go completely unnoticed in the eyes of some stubborn pessimists.

The objective of this study will be to report all the arguments found so far in favor of the existence of a proof of the theorem by Fermat himself. Although I have for a long time nourished a pronounced taste for mathematics and physics (ah! the discovery, in my youth, of Integrals, of (mechanical) Dynamics, of formulas so beautiful, so simple and so logical). I am not a mathematician, only an anonymous polymath, a small philosopher and above all a great curious person.

Many mathematicians, professionals and amateurs, have become passionate about this enigma, imagining a demonstration that would be "elementary" (short). Alas, this apparent simplicity veils unsuspected difficulties. For a long time, renowned mathematicians have been invaded by letters from Fermat admirers who submitted a proof that was of course always inexact (it is still sometimes the case but it is most often via the Internet that they make it known. In fact, there is a new pseudo-demonstration about every week on the net). This proliferation of letters has become so tedious that in 1908, Paul Wolfskehl created a prize of 100,000 marks to reward the first person who would found a proof of the great theorem. More or less eccentric demonstrations began to accumulate on the desk of Professor Edmund Landau, head of the Department of Mathematics at the University of Göttingen. He had been charged with examining all these proposals for proof. Their number was increasing so much that his personal work was suffering. He found a radical solution by having ready-made answer models printed in large quantities:

Dear…

Thank you for your manuscript on the proof of Fermat's Last Theorem. The first error is found: page…, line… This invalidates the demonstration.
Professor E.M. Landau

He then asked his students to fill in the blanks. The letters kept coming. The atmosphere around this famous theorem became more and more confused, and it seems that the collective unconscious and the 'group effect' at work in the highest mathematical circles decided that it was necessary to stop it and to disqualify, by all possible means, Fermat and his great theorem. The influx of these wrong demonstrations also served as a pretext for Fermat's detractors to deny even more fiercely that Fermat had proved his most famous conjecture. Let us pay tribute to all the scholars who have shown restraint and wisdom. The discovery of a proof by Wiles in 1994 aroused enthusiasm around the world, sometimes mixed with a little sadness: to prove a very elementary statement it had been necessary to write an entire treatise on mathematics, of unheard-of difficulty. Never a very academic mathematician could have imagined that Fermat (whose mischievous pedagogy was however well known) would have inserted in his second observation, written in Latin, all that was necessary to decipher it.

Our mathematicians no longer know how to reason soundly about primordial concepts, as they never have been forced to do so since their predecessors, more and more, have skipped stages. They are completely in our time, a materialistic era. The mind is increasingly cluttered with complicated thoughts, just like their way of seeking. The abstraction in the simple has become inaccessible to them, the pure spirituality, its beauty, is definitely lost. To reason they now resort to more and more mathematical symbols, more and more complex formulas, their thought is based on this complexity instead of being a pure thought.

Let us quote Alexander Grothendieck in Récoltes et Semailles (Harvesting and Sowing - Reflections and testimony on a past of mathematician):
"Our spirits are saturated from a motley “knowledge”, a tangle of fears and laziness, of cravings and interdicts; of informations all comers and explanations push-button closed space where come to pile up informations; cravings and fears without ever engulfing the wind of the broad. Exception made of how to make a knowledge routine, it would seem that the principal role of this “knowledge” is of evacuating a living perception, a recognition of things from this world. Its effect is mainly the one of an immense inertia, a burden often crushing.

Sometimes any of us discovers this or that. Sometimes he then rediscovers in his own life, with wonder, what it is to discover. Everyone has within them all it takes to discover all that attracts them in this wide world, including this wonderful capacity within them - the simplest, most obvious thing in the world! (One thing however that many have forgotten, as we forgot to sing, or to breathe like a child breathes…). Anyone can rediscover what discovery and creation is, and no one can invent it. They've been here before us, and are what they are."

Mathematics, especially Fermat's mathematics, is also philosophy. This study calls on many disciplines: mathematics, history of mathematics, philosophy (including philosophical logic), psychology, sociology, linguistics, pedagogy, didactics. The question considered can also help to understand our times. This research, initiated in January 2019 still does not seem to want to end. Apprehending the psychology of such a character to try to find all that he wanted to signify by his literary tricks is an endless work. It is only over the course of these discoveries (we go from surprise to surprise) and at the cost of long meditations that we can progress. This work was difficult at the beginning, because the collective imagination is there, which constantly recalls the final judgment that great scholars have made against Pierre de Fermat.

The story of Fermat's Last theorem begins around 1638. Fermat is maybe 30 years old. We can better understand his unquenchable thirst for knowledge by considering that he lives at a time when, without denying anything of the knowledge of the Ancients, on the contrary by admiring them, scientists focus on this knowledge to better move forward. They are polymath, everything is worthy of interest. Fermat is one of these men, humanist, scholar, philologist, he knows Greek and Italian, writes French, Latin and Spanish verses. A native of Beaumont-de-Lomagne in Tarn-et-Garonne, he moved first in Bordeaux, then in Toulouse, making a career in the judiciary where he carried out his task in an exemplary manner. When he discovered the arithmetic of the Ancients, he saw such intelligence, such stimulation for the mind, that being satisfied with a paid activity having above all the advantage of ensuring his subsistence is not even a question to ask. He sees in the study of numbers the royal way to contemplate the mysteries of Nature. His overflowing enthusiasm has found a way to express himself, his path is clear. Thanks to him, knowledge will be able to increase and spread. The science of numbers is not his only passion, Latin, the language of scholars, has no secrets for him. "He was shaped by Latin rigor and intelligence: it was on this soil that his prodigious genius in mathematics could flourish." He is very religious (see his Latin poem "Submission to God or the agony of Christ", dedicated to Jean-Louis GUEZ de BALZAC), discreet in life, and although he was a genius,"the tallest man in the world" according to Blaise Pascal, little is known about his life. By his correspondence, we know some of his most beautiful demonstrations, one of the most remarkable is when he shows that the number 26 is the only one to be between a square and a cube: 25 (5x5), and 27 (3x3x3).

One day, while he is contemplating the beauty of the Pythagorean theorem (a² = b² + c²), he wonders: could we add something to the subject, something that no one would never have dared to think of? In the formula, the exponent is the number 2, the only number which squared is equal to its double (2² = 2 + 2). Fermat might have thought that this property gave to it some particular properties, and suddenly, an idea took hold of him, an idea which was going to profoundly alter the course of mathematics in centuries to come. The unthinkable happens, he replaces the exponent 2 by a 3. Could the equality still exist for some cases, by carefully choosing the values of a, b and c? We can already see the range of its ambitions. A priori it didn't seem like it was possible, he could always get very close, sometimes even to a unit, but finding a solution seemed impossible. The number 2, a mathematical monster, seems to suggest it (to the unity, we added an unity to make a double unity, a philosophically blasphemous manipulation - or wonderfully creative? 2 is not only the first of the prime numbers, it is also the only even prime number. For Fermat, trying to prove the impossibility of his equality would be a formidable challenge, and that's all he needs. Certainly he realizes that it would be easier to test his method first with a 4 exponent, the square of 2, a number that seems to taunt all his followers. He establishes a method he calls "infinite descent" – or indefinite descent –, a reductio ad absurdum argument coupled with an inductive reasoning, and this is extremely effective. The method works perfectly with 4. It works harder with a 3 exponent. In September 1636 Fermat will begin to seriously excite the curiosity of his correspondents in a letter to Mersenne for Sainte-Croix, with the chalenge "to find two fourth powers the sum of which is a fourth power and two cubes the sum of which is a cube."

From exponent 5 and up to infinity, he quickly understands that his method doesn't work any more. He has to find another way, which will certainly have nothing to do with the descent. In 1670, five years after his death, in a short and provocative “OBSERVATIO” that he wrote in Latin, kept jealously secret during his lifetime, but that his son Clément-Samuel makes known, he claims to have really exposed a certainly astonishing explanation [of the general theorem] that the margin, too narrow, could not contain”. To this observation Samuel has added 47 other ones, all of which are inserted in the appropriate places in Book VI of Arithmetica by the Greek mathematician Diophantus which was published in 1621, and where Bachet de Méziriac had added a translation from Greek to Latin. In 1670, therefore, we have a new Arithmetica, that has been slightly increased, but will be very important for the future. The observation in question relates to question VIII of Arithmetica, it's the second of the 48 ones and it stands out notably from the others. We'll come back to that.

In the old days no one was solicited from the earliest age by all the vanities that now clutter our children's minds. Great intelligences were thus able to attain great knowledge by penetrating the essence of things. Socrates and Euclid were some of these great men. Much later and in the same century, Pascal, Leibnitz, Fermat of course, who was a famous example in the theory of numbers, building powerful reasonings with sometimes the only use of words.

Like Pythagoras, Fermat knows that when man has posed 1, then 2, everything is already posed, the uniqueness, the plurality of the world. However, something must have especially pleased him with this first plural number, to make Pythagore's theorem totally unique by its power, its singularity, by imagining a much more plural conjecture. First of all it was necessary to consider the most important properties of the first integer following the unit, the doubled unit, the first even number. In a second step, consider a new conjecture with a property that is related to the first one, but calling this time the infinity of integers (note that 1, the unit number, is not directly present in the "comparison", it is "apart"). Weighing the pros and cons seemed a priori a gigantic challenge. Certainly very quickly Fermat realized that the two the scale pans of the balance could never be at the same height, a ‘mise en abyme’ is impossible. He will therefore had to prove it.

The question of the Last theorem is much more than an arithmetic question. Its history is like a profound symbol of the historiography of Mathematics. Taking up the idea of Eric Temple Bell we are certain that civilization will die out before our mathematicians can understand Fermat's explanation.

‘’I have found’’, or ‘’I have discovered’’, is not said in Latin detexi, but inveni. See Invenio (inveni, inventum, invenire) in the Gaffiot dictionary: to find. See also ‘’inventio’’: action of discovering; discovery.

detexi, from the verb detego: to unveil, to lay bare (on Gaffiot Dictionary). In french language, detego, the infinitive form of detexi, can be translated by “découvrir”, but only in the sense of “to remove which covers”. The thing is that this french word “découvrir” has another meaning: to discover (to find). The first french mathematicians who had read Fermat's observation made a wrong translation and wrote: “I have found” (or discovered). This practice has been easily perpetuated until today (habits die hard). These constant wrong translations can easily be explained, bacause only the principle of the theorem, that was perfectly stated, seemed important for mathematicians.
sane: really, truly;
demonstratio: action of showing, demonstrating, describing (on Gaffiot Dictionary).

Here is the literal translation, the most exact that one can find, of the famous observation of Pierre de Fermat stating his theorem:
"Now, would it be a cube into two cubes, or a fourth power into two fourth powers, or in general up to infinity, no power higher than the square can be divided into two like powers, the surprising [or admirable] explanation of which I have CERTAINLY UNVEILED. This narrow margin could not contain it."

We'll see later that another valid interpretation of this enigmatic observation is possible. Fruit of a very elaborated encryption/decryption, it becomes obvious once you have it under your eyes. This decryption was carried out by Mr. Roland Franquart in 2008.

Here is the only official translation, which is by Émile Brassinne, in his work Précis des travaux mathématiques de P. Fermat et de l'Arithmétique de Diophante (Toulouse, 1853). This official translation is quoted by Serge Coquerand in his book À la (re)découverte des dix livres de l'arithmétique de Diophante, as well as by Bertrand Hauchecorne, who expresses it in the France Culture program Pierre de Fermat l'énigmatique (at 19 '25' '):

“To separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers, is an impossible thing, and I have truly found the admirable demonstration of it. The too narrow margin could not contain it.” The order of the words is strictly exact (adverb before verb), the only wrong translation concerns the word detexi, unfortunately translated by I have found (‘I have found’ would have been: inveni, in Latin).

We still read, in books or on the internet, inexact translations of this kind: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have ~~discovered~~ a truly marvelous proof of this, which this margin is too narrow to contain.

This translation was cited in 1950 by Jean Itard (1902-1979), who was a French mathematics teacher and a historian contemptuous of Fermat: "Never Fermat has been in possession of a proof of his Grand Theorem for a exponent greater or equal than to five." It's the most derogatory comment that one can find about the greatest mathematical genius of the seventeenth century. The beginning of the sentence is peremptory and violent: “Never Fermat...”. He goes on and puts two capital letters to "Last Theorem", a radical way to demolish a... ‘capital’ theorem. As Jean Itard was not versed in Latin he certainly relied on one of these wrong translations. It's probable that Fermat had arranged for his note, written in Latin, would easily be badly translated, since this would suit the commentators who would take him for a braggart, an "amateur".

The spread of these incorrect translations has had three effects:
– Tormenting scholars for 324 years (1994 - 1670 = 324) and not 358 years (often rounded up to 350) as is often written.
– Making it possible that Pierre de Fermat's greatest challenge - to find and fully understand the admirable demonstration he "really laid bare" by probing "the mysteries of science of numbers" - has still not been met in 350 years (2020 - 1670 = 350).
– Enabling Wiles’ complex work to bring considerable progress in twentieth-century mathematics.

“During the seventies the discouraging effect of these 350 years of relative failure was such that it was fashionable to say that Fermat's assertion was not general enough to be considered significant or that it was either undemonstrable or wrong. But in the space of about thirty years the mathematical community has radically changed its perception of the question, going from a more or less courteous disinterest to the most lively enthusiasm! People suddenly began to believe in the veracity of Fermat's assertion around 1985 and this disposition of mind was a powerful stimulus for the edification of the difficult theories that led to his demonstration.” Yves Hellegouarch, 16/06/2000, text taken from the 168th lecture of the "Université de tous les savoirs" (pdf).

The genesis of this research

The first reading (around 1997) that made me interested in this problem was Simon Singh's famous popularization book Fermat's Last Theorem a reading suggested to me by a friend of mine who was a student in mathematics. I felt that there was something exciting for me there^[1]. Charles Baudelaire says in one of his little prose poems: "I passionately love the mystery because I always hope to unravel it." ("J’aime passionnément le mystère, parce que j’ai toujours l’espoir de le débrouiller"). I too have this passion, taken to a high degree. Often a mystery is considered unsolvable, for the very reason that should make it look "easy" to solve. Concerning this theorem, it is perhaps only around 2010 that I discovered my first clue, a strange formulation of Fermat, a little ambiguous, in his last letter to Carcavi where he alludes to the famous wrong conjecture. I sensed that I was in for a tough game (it was Fermat!). He was certainly hiding a lot of things, but at the same time he must have left many clues since he seemed to have already left one. And I understood then that it was not answers that I had to look for, but questions, good questions. I found these first:

Was Fermat ingenious enough to have found, by a side step, a "simple" and very short proof, very deep and complex, also very far from the methods of traditional calculus?
Why is everything so strange about this theorem? Did Fermat have one or more reasons to be so mysterious? For example:
Why is his most famous observation written in the mode of a joke (much more than some others)?
Why does he write the incorrect conjecture about numbers of the form 2²ⁿ + 1 in an ambiguous form the last time he formulates it, a form that would lead those who would absolutely refuse to trust him, to believe that he thought this conjecture was exact and that therefore he was unreliable on everything else?
How is it also that his Arithmetica - the foundation of all his work, where his 48 observations were supposedly noted, and which became, especially after the discovery of the statement of his great theorem, a historical document of considerable value - has disappeared? Why did his son Clement Samuel not keep it? Did he have a good reason?
Shouldn't we analyse in depth everything he writes that revolves around the theorem?
Is he an honest man? Does he have very good morals? He certainly does. So why not start by trusting him?
Corollary: why did some commentators not trust him? Why, on the contrary, did they belittle him?
And why have his commentators never felt the need (to my knowledge) to ask themselves these questions?

With simple common sense, in a fine perception of the things, an objective approach devoid of any prejudice, as we progress in the research, our discoveries bring us a priceless satisfaction, we receive a wonderful gift. One who has the spirit of discernment knows how to think with simplicity, confidence, humility, creative imagination, boldness and rigorous analysis, all faculties needed to solve an enigma. I believe that solving the most important enigmas (whether the notion of infinity is an essential part of the mystery, or it is absent) is always possible. But in this case, even though I searched a lot, almost always with the same enthusiasm, I only found a few scattered clues at first. It's true that in putting them together they comforted me a lot in my initial intuition, and even if they did not lead to anything concrete, they already constituted, after an objective overview of the general context several times reiterated (in which I included not only Fermat's words, but also those of all his detractors), a good start to the analysis. I had to wait a dozen years before receiving in 2009 a private message via French wikipedia, from an amateur mathematician (Roland Franquart) that would completely unblock the situation. We called each other and I think we chatted for over an hour. We then worked on a dedicated blog where a doctoral student came. Then I continued to try to make the fr.wikipedia article on the theorem a little more reliable without achieving much, since a very strong opposition, even for the simplest details, prevented me from doing so. Fortunately when I returned to fr.wikipedia in 2013 after a long absence I was immediately welcomed and encouraged by Catherine Goldstein. She told me that she considered the two articles "Fermat's Last Theorem" and "Pierre de Fermat" as indigent, and she no longer venture into this minefield. Finally giving up myself to try to improve a little more these two articles and some others, I left fr.wikipedia completely disillusioned and I continued my research deeper. I had no idea then that by working alone and with a free spirit I would make a lot of progress through more and more astonishing and numerous discoveries that, after Roland Franquart, I would also make. I must say, to be fair, that without his discoveries I would never have found anything new, all this research could not have been done. Honor to whom honour is due.

Around 2006, after consulting fr.wikipedia about this theorem, I immediately saw that of all the arguments put forward by Fermat's detractors, taken up by Wikipedians, none was proved, none was seriously admissible. Many contemporary scientists, from all disciplines, reason with a form of magical thinking and show condescension or even open contempt towards the Ancients. This condescension is part of the common mores of accomplished mathematicians. God knows if I am aware of how many people there are who have diplomas like so many certainties, people whose academic recognition confirms their blissful certainties. The question to be asked on seeing the astonishing way in which the article in question was written was "Why?" This was the first stone to be raised in order not to be contaminated by the ambient pessimism and to start on the right foot.

This jealous and exacerbated conformity with one-track thinking being obvious, I first wanted to list all the bad arguments, with their harmful consequences, that over centuries Fermat's commentators have imagined. Then, since he had deliver his ultimate challenge, and since I had quickly enough perceived his manners, and that I admired the man, it was necessary to do everything to take up this challenge. Not the mathematical challenge in itself, since I am not a mathematician, but the challenge of unearthing other secrets (other than those already discovered by Roland Franquart) that he could have concealed in his various writings related to the theorem and the famous wrong conjecture. The difficulty being that he never said more than necessary, and the best signals he sent were the most difficult to access. Thus was born this research, very laboriously at first. Trying to solve as exhaustively as possible this formidable enigma, which requires a thorough analysis of Fermat's psychology, of a lot of his writings, which also requires an awareness of his admirable sagacity, aroused a lot of enthusiasm in me, in an exciting research which gave me the greatest satisfaction. If I had been a mathematician, I would never have thought of searching with so much faith and perseverance for all these arguments to rehabilitate Pierre de Fermat and his last challenge, because I would have been prevented by prejudices and a conformist way of reasoning from going off the beaten track during these centuries which had led to an incredible scientific imposture.

The sophisticated techniques used by the contemporary mathematician require a long apprenticeship, a lot of work, and occupy all his time. His professional constraints do not allow him to devote any more time to a question that seems to him of so little interest. To unravel such an enigma, it is the singular pedagogue, the isolated combatant, who must be summoned. His weapon of choice is the challenge. But so that the mathematicians who would follow him would not be too angry, he must not challenge them too openly, so he found a new weapon, the delayed facetiousness, which he used in profusion. To have a chance of meeting this challenge, it was necessary to go directly to the source, to find and exploit the most exact, most faithful translation of OBSERVATIO II. Then, hoping that he had not left it at that, it was necessary to continue to search with unfailing obstinacy for all the other leads he might have left. It was long, full of pitfalls, often exhilarating.

"The historian must be able to hear everything." (Cicero)

The Ancients had managed to extract from a formless arithmetic gangue the main concepts without having all the algebraic symbolism available today. Pierre de Fermat, like his contemporaries but to a higher degree, mastered the art of getting around the difficulties that those who came after him would face, to the point of being able to do without the many mathematical tools that would be discovered later. We now find that the primordial concepts that these Ancients developed are obvious. Up until the last century, and even sometimes even today, this obviousness generated in some scientists, when they had to scrap with Pierre de Fermat, their master however, a guilty arrogance.

Going against all the negative judgments that have been made against it, is not easy, two things help to keep enthusiasm and confidence intact.

We know on the one hand that he had very little time to satisfy his passion for numbers. Only by keeping with him the vast majority of his inventions as he made them, could he preserve his tranquility and exploit his full creative potential. If he had started to write complete demonstrations of his inventions, understanding would have been very difficult for the other mathematicians, fussy minds would have wasted his time with endless quibbling. The wording of his challenges, which often contained only a few lines and could seem inappropriate on the part of a notable, also testified to this cruel lack of time.

While he claimed in his correspondence to have the proof of the particular case n = 4 of his great theorem, he does not tell us, in his Arithmetica, what this proof is. Nevertheless, he explicitly gives us the proof of “Fermat's right triangle theorem”, without telling us that this theorem has any relation to the case n = 4. However, the proof of this case is immediately deductible, and it is the only proof he reveals - in his 48 "observations" anyway. At first glance, it is incomprehensible because never showing false modesty, he never hesitated to put forward his abilities. For what reason then, if not to indicate that he placed a first tag there, and to show that we will have to focus on looking for others tags, better hidden. His assertions on the impossibility of the particular case n = 3, and especially his assertion of having really exposed the astonishing explanation of the general theorem. This seems to us to be the very first of the arguments in favor of a complete mastery, by Fermat, of the situation: he knows what he is talking about, he lets us know it. Furthermore it is certain that he does have the proof of the case n = 3, and while he has not ceased to talk about it in his letters, If he hasn't stopped talking about it in his letters, he doesn't give the demonstration. He arranges, as he has done in his letters, to reveal only the minimum. If he had renounced this principle, mathematicians would not have had to make any effort.

He also certainly had to take his revenge on the community of mathematicians ("Ah! They didn't want to take me seriously? It's a shame for them! I don't think of these stubborn minds anymore from now "). Did he also said to himself “It will be funny." ? In fact, some of his correspondents to whom he had submitted problems which they had been unable to resolve, had despised his work, deeming them totally useless (these problems proved later to be of considerable importance). He must have been contrite and annoyed and may be he wanted to punish them for their negligence. The nature of his character must have had something to do with it, he was known to be very humble, but he was perfectly aware of his strength, and false humility was extraneous to this Gascon. A complete proof of the particular case n = 3 of his great theorem will only be found two centuries later by Gauss, another great mathematician. Let's quote E.T. Bell: “Gauss discredited baseless assertions. […] A friend had asked him why he did not compete for the prize offered in 1816 by the French Academy of Sciences for a proof (or an invalidation) of Fermat’s Last Theorem. I admit, he replied, that Fermat's Theorem is an isolated proposition which is of very little interest to me, since I could easily find a multitude of propositions of the same kind, which no one could ever validate or invalidate.” Although he never said so explicitly, Gauss seemed to doubt that Fermat had proved his theorem.

On the other hand, some of his most important works are written in Latin, the language of the ellipse par excellence. Fermat being an expert in Latin, it was necessary to flush out as much as possible of his unsaid - written, but subtly hidden - due to: a) the concern for discretion in troubled times (he was a magistrate); b) lack of time; c) the very principle of the challenge, which was consistent with the previous two points; finally, d) his taste for pedagogy (which in turn agrees with the previous points).

When we study Fermat, there are two ways to proceed:
1) With a positive outlook: always remember that he is a great pedagogue, trust him, detect all the specious arguments put forward by his detractors, on the contrary look for the numerous clues that he leaves us, and all the good arguments (I counted fourteen important ones). The authors who have published a book on the great theorem have had the wisdom to remain fairly neutral.
2) With a very suspicious preconception: to underestimate him, not to trust his most laudable desire never to make our work easier. We then imagine multiple arguments to discredit him.

If one looks for the entire book Fermat is said to have devoted to the science of numbers, one will find much in the Arithmetica of 1670, which includes 48 very stimulating observations. The nugget that appears there is an astonishing joke. Never before, and never again, will we see a genius, even a universal one, deliver the demonstration of a powerful theorem in the form of a statement that leaves so much to be thought about: "I really have unveiled the most astonishing explanation that the too narrow margin would not contain".

Being a professional mathematician has advantages and disadvantages. Among the latter, one emerges: you practically only think about your work; your mind, consciously or unconsciously, is occupied with it day and night.As for the study of the Fermat case, many mathematicians and historians have studied it, but a real consensus has never been reached. Would you waste your time studying it? If you are a simple, confident, objective and daring amateur, the problem is different, you first notice that none of the arguments put forward by the skeptical commentators is really serious. Their assembly is even less serious and the whole edifice collapses if you remove the worst ones:
1) Fermat had already made a mistake with numbers of the form 2^2ⁿ+1 (Fermat numbers) (we shall see later in what way and for what purpose this was a bluff on his part).
2) He must have made another mistake in his famous observation, he realized it later, but since these observations "were reserved for his own use" (...) he did not consider it useful to correct them.
3) He did not have our modern tools.

If you are just a very attentive amateur, you immediately see that there is something fishy going on. So you document yourself. For a long time if you are an enthusiast. A remark very quickly comes to your mind: these commentators seem to have started from the premise that Fermat could not have found a proof, then they looked for anything that could support them in this idea, aggregating their arguments in a single block to make it a quasi-certainty - if not more. You then wonder about the intellectual honesty of some scholars. The amateur that you are says to yourself: “All this is just smoke and mirrors”. Fermat, whose real profession was magistrate, always considered emulation as the best stimulant in arithmetical research. He tried everything, for 19 years he challenged 7 of his correspondents: to prove or disprove his false conjecture about Fermat's numbers. But none of them really followed up.

The attitude that one has, when faced with the "Last Theorem" (it sounds like the title of a novel, which it is indeed) depends on the a priori chosen at the beginning. If you choose the favorable one, you tell yourself that Fermat, pedagogue and facetious at the same time, and an honest man if any, should not have stopped there. You are then ready to search assiduously for all the clues he could have left us, leaving no stone unturned. Eric Temple Bell believed in a Fermat proof and thought that civilization would probably die out before the Last Theorem was solved. However, it was not totally excluded that the theorem would be proved one day by a very complex method: it was indeed the case, and other complex proofs will be discovered. To believe at any cost that Fermat could not find an empirical proof, therefore much shorter than the one of Wiles in 1994, is a blatant misjudgment of the science of numbers and more generally of the scientific method.

Mathematics and poesy, spirit of geometry and spirit of fineness

Blaise Pascal, in Les Pensées, distinguishes the spirit of geometry and the spirit of fineness^[2].

For a pure mathematician who is only a mathematician, the greatest evidence will always escape. I have read very few mathematicians who, apart from their mathematics, could be trusted. Only the mathematician who has kept his childhood spirit can reason clearly; he must be a poet, who never restrains his creative imagination. Let us quote Etienne Klein about Einstein: “This is perhaps what I admire the most about him. This ability he had to ask himself simple questions, childish questions, and to find answers to them elaborated with all the rigor of an adult brain.’’ Let us remember that Fermat wrote poetry (in several languages), likewise Giordano Bruno. Let us remember the unforgettable logician Lewis Caroll, author of 'Alice in Wonderland' and 'Through the Looking Glass'. Let us remember Jacques ROUBAUD, writer and mathematician, member of the Oulipo, a very spiritual Go player and poet, well known to mathematicians, who conveniently reconciles "the spirit of geometry and the spirit of finesse". Then let us notice that Catherine Goldstein, a researcher in mathematics and historian (who always said, contrary to a flock of knowing that the existence of a proof of Fermat's theorem by Fermat himself was not improbable), had for father a poet, Isidore Isou (1925-2007), who was also a painter, a novelist, a playwright, an economist…

According to the mathematician Jacques Hadamard, daydreaming and imagination have a big role in mathematical invention, it is often by imagining a new path that the greatest researchers have "seen" a solution that has hitherto been inaccessible (the word "theorem" comes from the ancient Greek θεώρημα – theorema in Latin): a proposition object of contemplation, of meditation. According to Gaston Bachelard, imagination confers above all the power to free us from the primary images provided by perception by deforming them, by changing them: "The fundamental word which corresponds to the imagination is not image, it is imaginary."(L’air et les songes. Paris, José Corti, p. 7).

In the following lines we will see how much the strategy that Fermat put in place to deliver his ultimate challenge, is not only a challenge to the imagination, but borders on a police enigma in which, while investigating, we find all the charm of a poetry. Edgar Allan Poe, poet and famous novelist precursor of the mystery novel, whose writings were translated by Charles Baudelaire, if he had been aware in his time of the discoveries made by Roland Franquart, would have been delighted to have to carry out a real investigation this time.

Mathematics are concerned with quantities and forms, they are not the whole thing. A reason only cultivated with algebraic logic is irrelevant. Only a reason governed by general logic is valid. Mathematicians have implicitly postulated that a purely algebraic truth must be a general truth. The confusion is so enormous, the error so gross, that one can only marvel at the unanimity with which it was accepted. Likewise, a mathematical axiom cannot be an axiom of a general truth. They also they decided (French mathematicians are the most guilty of this scientific sham) to apply the term ‘’analysis’’ to areas of their discipline, considering that the words derive their value from their application. Try, if you are not afraid of being slaughtered, to explain this to a pure mathematician who only reasons with his algebraic reason.

Fermat on the psychoanalyst’s couch

First link, the bluff: Numbers of the form 2^2ⁿ + 1

"It is what I find that brings me what I am looking for." (Pierre Soulages)^[3]
"At the heart of every difficulty lies an opportunity." (Albert Einstein)

Mathematicians agree that Fermat knew the method that shows whether numbers of the form 2^2ⁿ+1 (Fermat numbers) are prime or not. And they are not. One of the explanations given by his most deluded commentators about what Fermat thought about his false conjecture, is that he must have made a calculation error (...), but that he had not verified his assertion (...), whereas the verification is done immediately.

For almost a third of his life Fermat has always said he had no proof of this proposal and he urged all his correspondents, time after time, to come and help him. A period of 19 years was certainly not enough to allow this genius to verify a very simple calculation that takes only a few minutes for the case F₅ . A nice demonstration can be read on the blogdemaths site. Michael Mahoney, a professor of history and history of science, thought that Fermat would have been sure of the validity of the conjecture only up to F₄, and would have deduced that it must still be valid without even checking the F₅ case. One is amazed to see how easily educated people can sometimes write eminent nonsense.

Using prime numbers 74k+1, Fermat invented a very ingenious method to prove that 2³⁷ – 1 (ie 137,438,953,471) is divisible by 223. Using a similar method, divisors 64k + 1, he would have immediately shown, with the help of 4 short divisions, that F₅ is divisible by (64×10) + 1 (ie 641), therefore is not prime, therefore the conjecture is false. Moreover he did not join it to his 48 observations in connection with the Arithmetica of 1621, that his son transcribed on the Arithmetica of 1670 ("where all these propositions were later recognized as rigorously correct"). This absence reinforce the view that he always knew it was false. Let us also mention a letter from Fermat to Mersenne (dated April 7, 1643) in which one can read:

"You ask me whether the number 100,895,598,169 is prime or not, and a method of discovering, within a day, whether it is prime or composite. To this question I answer that this number is composed and is made of the product of these two: 898,423 and 112,303, which are prime." The methods that Gauss (1777-1855) would later propose in his Disquitiones arithmeticae to decompose a composite number into its prime factors could not have solved the problem proposed by Mersenne.

In her book Un théorème de Fermat et ses lecteurs Catherine Goldstein studies in particular Fermat's Observation XLV (its formulation, readings, etc.), which is the only complete proof of a theorem that he unveiled (Fermat's right triangle theorem). This proof implies the impossibility of case n=4. Overtime mathematicians have made different readings of this theorem. C.G. makes her own reading which has the advantage of answering "all the objections raised so far". On page 148, note 4, she notes that “important letters for researches on numbers do not appear in the VARIA OPERA MATHEMATICA^[4] (These various mathematical works are a selection of articles and letters). When Samuel published them after his father's death, as he did for the observations, he inserted a single letter evoking the wrong conjecture, the one addressed to Monsieur de ****. It's certain that it's still Frenicle de Bessy. A second letter then, about two months later: published by his son in 1679] as the letter to Carcavi in 1659”. The ambiguity of a passage of this letter made many commentators say that Fermat must have been mistaken in his conjecture). It is noted in these Varia Opera (Paul Tannery had noted it) that the first letter from Frenicle de Bessy misses as well. It is not the only one, apart from one, all Fermat's letters about the wrong conjecture are missing:
1) To Frenicle de Bessy, ‘August’ (?) 1640, where these words appear – the context has never been studied (see below) by the commentators of Fermat: “[...] but I have excluded so many divisors by infallible demonstrations [...]”. If we trust Fermat, we think that he formulates thus to stimulate Frenicle.
2) To Mersenne, Christmas 1640, “If I can once hold the basic reason that 3, 5, 17, etc. are prime numbers, it seems to me that I will find some very beautiful things in this matter, for I have already found some marvelous things which I will tell you, after I will have had your answer and the one from Mr Frenicle.” (always the same baits).
3) To Pascal, 29 August 1654: “and I confess that I have not yet been able to find it fully; I would not propose it to you to seek it, if I had overcome it. This proposition serves for the invention of the numbers which are at their aliquots in a given reason, whereupon I made considerable discoveries. We will talk about it another time. I am, [etc.].”
4) To Kenelm Digby, for John Wallis, 19 June 1658: “It remains to find a demonstration of this proposition, certainly beautiful but also very true.” (Letter XCVI in Œuvres de Fermat, t. 2, p. 402-405).
5) To Carcavi, August 1659, in a balance sheet for Christiaan Huygens, who had other points of interest, quite different. The formulation of this conjecture is very unusual for him:

“I then considered some questions which, although negative, do not remain to receive very great difficulty, the method to practice the [infinite] descent being quite different from the previous ones, as it will be easy to experience. These are the following:
- No cube is a sum of two cubes.
- There is only one square in integers which, added to the duple, gives a cube. Said square is 25.
- There are only two integer squares, which, increased by 4, make a cube. The so-called squares are 4 and 121.

- All the square powers of 2, added to 1, are prime numbers.
This last question is of a very subtle and ingenious research and, even though it is posed affirmatively, it is negative; because to say that a number is prime is to say that it cannot be divided by any number.”

Eventually it's this ambiguous formulation that will become after his death the most famous of his remarks on these Fermat numbers. Huygens was a young 30-year-old scientist and mathematician, the only one who could still follow him, but he did not follow up. The formulation of this last trial balloon was, however, very exciting:

In these letters he asked for help (!) from his six main correspondents. One after the other he tests, stimulates, encourages them to follow him in his work. What a motivation for them, to come to the aid of the great Fermat). But none of them will answer, except Frenicle.

This time Fermat has "considered" certain "questions". Fermat does not use, as he often does, the expression "proposals". Besides, the expression "negative question(s)" is not very correct; a question, formally speaking, is always a question. The wording of the whole paragraph and at the end the allusion to prime numbers (which cannot be divided by any other number) allows him to introduce the term "negative". Fermat, philologist, uses it in a testament-letter. Is he insinuating that the answer to the question is negative? It is our thesis, better, knowing very well, since the long years we spent with Fermat, his psychology, his facetious spirit and his deep taste for pedagogy, our certainty. The formulation: "This last question is a very subtle and ingenious research [...]" is admirable. This question that he asks us, the readers, is made even more intelligent by adding, for no apparent reason, the synonym "ingenious" to the adjective "subtle", which is an overlap. So let's continue to trust him by showing finesse, creativity and, like him, "let's consider the issue". The wording of the paragraph has a double meaning: the research he evokes is not only the arithmetical one concerning this proposition, it is above all a search for subtleties in everything he writes. It will be therefore advantageous to understand the beginning of the sentence in this way:
"This last question is of a very subtle and ingenious research in its formulation [...]".
As we have seen, it's absent in the 48 observations transcribed by his son on the Arithmetica of 1670, all of which were later proved correct.
It's also absent in the Varia opera.
The arrangement of singular formulations in the whole paragraph is of diabolical skill.
Concerning the first 3 questions, Fermat showed that these proposals were true. A detractor will therefore immediately be inclined to believe that Fermat thought he had shown that the last one was also true.
Let's note that in the letter to Mersenne of June 1640 (see below) where Fermat uses the same method, this time with the divisors of the 74k+1 form, his son did not put it in the Varia as well. Six important letters related to the wrong conjecture are therefore missing.
We claim that these 5 letters are a huge bluff, this now seems crystal-clear as we also saw at the beginning of this section, yet we have never read that a mathematician thought of this possibility, so difficult is it to counter the panurgism that has run throughout the centuries. Nor would the contemptuous have thought that the very serious magistrate Pierre de Fermat, who has often been said to boast a little too much about arithmetic (and yet this was part of his strategy of defiance), would have emboldened himself to such a bluff, running the risk of losing a good deal of his credibility - which is exactly what happened. Let's remember that he did not care about glory.
Not only does he seem to want to show us how much he would have liked to have found partners to accompany him in his arithmetical research (did he really believe in it himself?), but all these letters, which are absent from the Varia, as well as the sixth one that appears in it (see below), serve another purpose: they set the stage by making it easy for the reader to believe that, after all, Fermat is not such a serious mathematician. In this letter to Carcavi, while he certainly had great doubts about Huygens' response, he left posterity with a memorable first message that was intended to be ambiguous and would cause much discussion. Fermat never stopped playing with us, the game started in 1640 and will never stop intensifying. The climax was obviously the famous observation on his great theorem... which he was careful not to publish during his lifetime. A wink from beyond will be made 30 years later thanks to his son Samuel with the 48 observations, for eventual confident and attentive readers.

"Sometimes, by commenting on some often confused impressions, perhaps on this or that particularly obscure and confusing passage, I managed, as I was writing, to penetrate more and more into the meaning of a text that had seemed to me to be hermetic. [...] As the days and weeks went by, I realized that the simple fact of copying in extenso a particular passage from the text I was examining, astonishingly altered my relationship to that passage, in the sense of an opening to the understanding of its true meaning." Alexander GROTHENDIECK, "Récoltes et semailles", p. 428.

"I even believe that the sudden onset of such a feeling [obvious] is more or less common in all discovery work, at times when it suddenly leads to new, great or new understanding. small. I have experienced this over and over again throughout my life as a mathematician. And it is the most crucial, the most fundamental things, when they are finally grasped, that are the most striking because of their obviousness; those which we say to ourselves afterwards are "gouged out" - to the point where we find ourselves amazed that neither oneself nor anyone else has thought about them before and for a long time. This same astonishment, I encountered it again, and just as much, in the work of meditation - this work of the discovery of oneself which came, little by little, to merge almost with the work on my dreams.
People tend to ignore it, that sense of obviousness that so often accompanies the act of creating and the emergence of what is new. Often we even suppress the knowledge of what may seem, in terms of received ideas, a strange paradox." Alexander GROTHENDIECK, La Clef des songes, p. 24.

For Pierre de Fermat geometry and arithmetic are both a passion, a work and a game (remember that he enjoyed working on magic squares). He uses Latin a lot, its rigor and conciseness correspond perfectly to the requirements of mathematics. In fact derogate from the precise rules of this language allows to play with words, the use of "the enigmatic ellipse or encryption" (Ludivine Goupillaud) being the most remarkable example. In this testament-letter he makes a translation from Latin to French and for the first and only time he uses the process of encryption in a text written in Latin. If you read between the lines: "To understand the ins and outs of this testament-letter, you will not only have to make an objective reading of it, you will also have to submit it to a rigorous analysis, since it is, in fact, the result of a very ingenious research. In turn, you will have to submit to a very subtle research." His detractors, by distorting his words, will doubt his competence and will use this letter as an argument to deny that he was able, with his own tools, to find a proof to his Last theorem. His supporters will rejoice in discovering these subtleties, which, if they are not as decisive as the encryption of his most famous observation (see below) are sublime as well.

(C.M.) When Samuel published them after his father's death, as he did for the observations, he inserted a single letter evoking the inexact conjecture, the one addressed to Monsieur de ****. It's certain that it's still Frenicle de Bessy. A second letter then, about two months later:

6) 18 October 1640: "But to be perfectly honest I must say (because in advance I warn you that, as I may not ascribe to myself more than I know, I say with the same frankness all that I do not know) that I have not yet been able to demonstrate the exclusion of all divisors in this beautiful proposition that I sent to you and that you confirmed to me, concerning the numbers 3, 5, 17, 257, 65537, etc. Because, although I reduce the exclusion to most numbers, and as I have even probable reasons for the rest, I have not yet been able to demonstrate the truth of this proposition, that, however, I don't doubt now, as I didn't doubt before. If you have got the undeniable proof, I will be much obliged to you, if you communicate it to me; because, after that, nothing will stop me in these matters." (!) The previous expression by infallible demonstrations is softened: two months only after his first letter to Frenicle, it seems that Fermat wants to reassure him about the difficulty of the proposal.

So Samuel omits all the formulations (in particular the last that gave rise to the controversy), to keep only this formulation clear, unambiguous, in an official document, since it's in a published book where his father's words appear: "[...] for I warn you in advance that, as I am not able to attribute more to myself than I know, I say with the same frankness what I do not know [...]" [1] (place the mouse pointer at the top, then in the black band that appears, enter the page number 181, here we access the real pages 162 to 164 of the book, Fermat's complete letter). Samuel's choice precisely for this letter, in which his father says he is always honest, no doubt wants us to understand that Fermat's observation on his great theorem must be seriously considered. His commentators have never wondered, to our knowledge, why Samuel Fermat published this single wording. On the contrary, at every opportunity they have found they climbed on their dewclaws. The last wording intended for Huygens, and over all the extreme difficulty to prove the general theorem, were for many of them arguments to consider that Fermat too much committed himself when he claimed to have found a proof with his own tools. This letter for has been his last attempt to find someone to share with. One can easily forgive a farmyard that was too excited (the laying of eggs was related) to think calmly. The "optimists" (the word is mostly used by the detractors or by those who do not pronounce themselves, I prefer for my part the expression "realistic people" (or objective, or lucid, or honest), the lucid people therefore, will say to themselves that this narrow labyrinth, where the beacons never cease to let themselves be discovered by adding one to the other as we advance along a path bristling with traps to guide us towards the goal of the hike, cannot be the fruit of chance. We are certain that Pierre de Fermat informed Samuel very precisely what he would have to do, to complete his father's "Great Work".

To Frenicle he writes "after that, nothing will stop me in these matters", but should we take this statement literally? Isn't it there above all to sharpen Frenicle's curiosity? Since if Frenicle would have been able to find the counterexample F₅ , Fermat would have found the ideal partner, their future exchanges could have been the subject of jousts and exchanges that would have considerably enriched historiography.

One can read in Fermat by Tannery, p.199 that he had used the argument of numbers of the form 74k+1:

7) Letter to Mersenne, June (?) 1640. "In addition, you or I have equivocated by a few characters the number that I had believed to be perfect, which you will easily know, since I leased 137,438,953,471^(*) for its radical, which I have since found, however, with the Abstract taken from the 3rd proposition, to be divisible by 223; what I knew at the second division I made, since the exponent of the said radical being 37, of which the double is 74, I began my divisions by 149, greater that unity than the double of 74; then, continuing by 223, greater than unity than the triple of 74, I found that the said radical is a multiple of 223."
"From these Abstracts I can already see a great number of others dawn, Et mi par di vedere un gran lume⁽^**).
I will tell you one day about my progress, if Mr. Frenicle does not come to the rescue and abbreviate by this means my search for abstracts. In any case, I beg you to see to it that Mr. de Roberval joins his work to mine, since I am in a hurry with many occupations that leave me very little time to do these things. I am (etc.)."
^(*) M37, Mersenne number which is not prime.

^(**) Translation from Occitan: "And it seems to me I see a great light."

From these few letters, as in all his correspondence, we notice two things that we might find contradictory.

Fermat never ceases to marvel (and rightly so) at his most beautiful finds. He praises their importance so much that some of his correspondents have turned him into a braggart, a label that has sometimes remained with him.
He has never sought glory. The only work he published was under the pseudonym "M.P.E.A.S.," the meaning of which was long ignored (see below).

We can easily explain the first point, as Fermat himself says his only goal is to make science progress, he feels an intense joy in it, as at the magnificence of his discoveries; his motivation is extreme.
The second point is even easier to explain.
a) He is known to be honest and modest by those who have known how to appreciate him. Descartes who was very jealous, and some others who could not compete were of a completely opposite opinion.
b) He is very busy with his career as a magistrate. Would he have been proud that fame would have been detrimental to his peace of mind, and time was precious to him as he had an unquenchable thirst for "admirable" discoveries.
c) The era is politically troubled, to put himself forward would have been detrimental to his career as a magistrate.

Last Problem

“The enigma is the infinite power of the known, it is what pushes the known towards its infinity, towards its thirst for knowledge, [...] it is a place of future adequacy. [...] Imagination opens onto creation and ethics.’’ (Cynthia Fleury, Métaphysique de l'imagination).

Over the centuries and their discoveries, mathematicians have become more and more sure of themselves, sometimes imbued with their knowledge. This pride of our profession (which we all have, and which is human), as well as a rationality with blinkers, sometimes take precedence over the creative imagination, restraining it. Let us borrow the words of Jacques Roubaud: "The followers of the followers [...] no longer know anything about what motivated the founders [...]. They think they know everything there is to know, right from the beginning.

The rigorous analysis of Fermat's second “OBSERVATIO” (question VIII of the 1670 Arithmetica), the study of his works, his correspondence, his life, and above all his psychology, is an unfailing subject of meditation. His Great Work consists essentially of :

– forty-seven observations noted ''OBSERVATIO D.P. F.'',

– one observation noted ''OBSERVATIO DOMINI PETRI DE FERMAT'' (the famous note).

These 48 observations, which would fit in a few pages, were added by his son Clément-Samuel to the 1621 edition of the Arithmetica, to compose the 1670 Arithmetica. Here is a new book that has contributed enormously to knowledge, a book whose "prologue", by Diophantus, is much longer than Fermat's text. Over time, this 17th century observation was very roughly translated into different versions which mathematicians have always relied on: the "theorem" being perfectly stated, they were content with it. They would never have imagined that Fermat's explanation was right before their eyes! It must be admitted that since Fermat was known to be very stingy with his demonstrations, it was very difficult to imagine that for his greatest theorem, he would have made our work easier... The Latin note itself was often badly transcribed, we know one whose first word has been transformed into Cubem: “Let us sleep! » We have not yet seen a translation of Cubum autem in duos cubos by "but I sleep with both elbows on the table" but a very untalented or joking student could well have done it...

Here is again the exact translation, Fermat intends it for the serious and honest researcher. We will see later that a second interpretation of this enigmatic note, evident after the decoding carried out by Roland Franquart in 2008, is possible. The evidence of this second interpretation is further accentuated when we look at a very particular version of Arithmetica (see below) that I discovered in 2017 after painstaking research on the internet. We will also see that the two interpretations each have a utility.

"Now, would it be a cube into two cubes, or a fourth power into two fourth powers, or in general up to infinity, no power higher than the square can be divided into two like powers, the surprising [or admirable] explanation of which I have CERTAINLY UNVEILED (or LAID BARE). This narrow margin could not contain it."

In modern terms: "x, y, z being positive integers, xⁿ + yⁿ = zⁿ, is impossible for any value of n (integer) greater than 2."

The translation that one usually encounters contains two major errors in the most crucial part of the text.

— First error: “I found a really marvelous demonstration of it. Fermat wrote detexi, not inveni (verb invenio: to find, to discover).
— Second error: Since Fermat placed the adverb sane (“really”, “certainly”) before the verb detexi (I have laid bare, unveiled, uncovered), it is therefore to the verb that the adverb relates: "I really have unveiled an admirable demonstration of it".

The mathematicians who mistranslated Fermat's note would have liked to make this conjecture a joke that they would not have done differently. By translating: "I have discovered (or found) a truly marvelous demonstration of it that the margin is too narrow to contain", they made him, by a magnificent Freudian slip, and definitively, a boaster, an amateur who claims a true thing without being able to prove it. And by this wrong translation they have further accentuated the mysterious aspect of this observation, preventing it from being seriously studied.

‘Lack of space’

(C.M., Jean Rousseau, Laurent Hua). Keeping one's own demonstrations to oneself was a common practice in Fermat's time, and how many of these demonstrations were not found only after the death of their authors? Can we believe that it was only for lack of time, and even less for lack of space in the margin, that he jealously kept his “marvelous” demonstrations to himself? Let us imagine for a moment that he had managed to write the entirety of each of his 48 observations in the margins. The longest of these would have had to be divided into several parts and spread over pages that were not directly related. Observation VII, for example, contains 697 words spread over 24 paragraphs.
1) As we have seen, these 48 OBSERVations were obviously written for the reader. Moreover, the style is as perfect as in his letters, are these the ways of writing a reminder?
2) They constitute the most formidable contribution to seventeenth-century mathematics.
What seems likely is that :
- Fermat wrote the less lengthy of these observations in the margins, and not in such an elaborate, literary form as we know them today.
- Toujours dans les marges, il a brièvement résumé les plus longues. Pour ensuite les formuler d’une manière très élaborée, très élégante à l'intention des lecteurs, soit sur des feuilles volantes assemblées ensuite entre elles, soit sur un livret. Avant d'être transcrites par Samuel sur l'Arithmetica de 1670. Parmi celles-ci la note très courte sur le grand théorème, dont le codage a dû nécessiter un travail intense, et dont la typographie «trafiquée» deux fois dans deux éditions différentes sur le même mot detexi, devait être parfaitement rendue pour que Samuel la transcrive exactement.

In the margins, he briefly summarized the longer ones. He then formulated them in a very elaborate and elegant way for the readers, either on loose sheets of paper assembled together, or in a booklet, before being transcribed by Samuel on the Arithmetica 1670. Among these is the very short note on the great theorem, the coding of which must have required a great deal of work, and whose typography, 'doctored' twice in two different editions on the same word detexi, had to be perfectly rendered for Samuel to transcribe it exactly.
- It is possible that Fermat also inserted information in the margins that was reserved exclusively for Samuel.
- Since Fermat wrote brief observations in the margins, Samuel did not even have to make a pious lie when he reports that his father inserted notes in the margins.

In our thesis Samuel is therefore "in the secret of the gods" (i.e. Fermat passed on to his son all the necessary instructions). He does not keep the “Diophantus” after having collected it from the printer, even though all of his father's observations were supposed to be in it. So where did this Arithmetica go? This is what Samuel de Fermat wrote in the preface to the 1670 edition: "Illas [observationes] Parens meus quasi aliud agens et ad altiora festinans margini variis in locis apposuit, præsetim ad quatuor vltimos libros." ("These remarks my father noted in the margin in various places, especially in the last four books, as if he were doing something else and was anxious to achieve higher goals."). The clarification "especially in the last four books" was not really useful since it is obvious. It is justified in the sense that it is in these Books III to VI that most of the 48 observations (45 out of 48) appear, but it is precisely among them that we find the longest.^[5]. On the other hand, the first three observations are very short and would have fitted in perfectly (I don't know if Samuel left us a clue here, but it is possible).

Samuel obviously knows that if the 48 observations had been written entirely in the margins, the work would have acquired considerable historical (and commercial) value. No commentator on Fermat has questioned the strange disappearance of such an important work. It has not aroused any curiosity. In our opinion, none of them has had the slightest inclination to try to understand it, to see it as an additional argument in support of a proof held by Fermat. Catherine Goldstein, without dwelling on the subject, uses the conditional and writes: "[...] observations that he would have written in the margin. If Fermat gave his son precise instructions, either in a booklet or on loose paper, on how to record this note, which was so important to him, in three different versions of the Arithmetica. three different versions of the Arithmetica, then these instructions perfectly justify the disappearance of the work, which Samuel was forced to destroy so that his father's plan could be fully implemented. We see no other explanation for the disappearance of the book. It is also clear to us that Fermat had asked Samuel not to make the 48 observations known until after his death. Pierre and Samuel, here is a very noble binomial that has well deserved its particle: Pierre, a man of heart, integrity, both humble, ambitious and daring, incisive in his challenges, "lazy" he says of himself, but rather extremely busy. Samuel, also a humanist, a devoted passer-by, he knows where he comes from, he knows where he is going, a well-directed vector in short, a worthy heir to his father.

Only a few very rare works were found in Fermat's library, including an Arithmetica by Diophantus. It was to be expected that it would not be a copy of the Arithemica similar to the one that had inspired Fermat and that it would not contain the 48 observations, as this would have been too blatant an indication that Fermat father and son had left to posterity. Not wanting to overlook anything, I called upon the good offices of Madame Marielle Mouranche, Curator of Libraries, responsible for ancient books at the University of Toulouse, who confirmed that the Arithmetica that had been found was not an edition of 1621, but another, published in Basel in 1575 [2] and commented on by five people, including Fermat.

The style of the 48 observations

(CM, Jean Rousseau, Laurent Hua). It's crystal clear that the style used by Fermat in his 48 observations, the care with which they were written, their elegance, suppose the presence of a reader. Moreover, why should he have explained to himself that he really laid bare a marvelous demonstration that the too narrow margin can not contain?.
Yet historian Jean Itard wrote: “These remarks were reserved for its sole use.” Likewise, after Andrew Wiles's discovery in 1995, Winfried Scharlau would like we believe it. A baroque argument is also advanced to deny a proof of the theorem: “since [Fermat] did not know our modern tools”. A naive person never wonders what could encourage many scientists, supposed to have a logical mind, to erect a stack of fallacious arguments to better discredit a genius, whereas these “scientists have at least one more prejudgment than the ignorants, that one of believing exempt of it. It is this prejudgment, by which they fight those of others, which renders incurable, to them, he disease of prejudgments”. (Auguste Guyard, Quintessences, 1847). Some urban legends die hard, especially if they are maintained by experts.
(Tannery) The title OBSERVATIO DOMINI PETRI DE FERMAT is the only one, among 48 observations, to be fully spelled out, not abbreviated OBSERVATIO D.P. F.; this suggests to observe minutely this OBSERVATIO. Why ? Would it be so important for the science of numbers? So, why don't we have a demonstration immediately understandable, even very concise?

It is hard to believe that the three different spellings of the same crucial word detexi on the same observation, in three different versions of the 1670 edition, if they had been accidents, would have escaped his son Samuel, who worked so diligently to make his father's work known. The same is true of the exaggeratedly highlighted dot which follows the word in question in all three versions. Should we also take for incredible coincidences all the curiosities that we discover in this observation (there are 9) when we analyse it in depth? On the one hand, we realize that these curiosities, when studied and put together, become interdependent by forming a very coherent whole, and on the other hand, that the specious arguments put forward by Fermat's detractors are often also put together, but without any link between them this time and with the sole aim of arguing that Fermat could never have found a proof: Fermat is making a lot of advances, or is boasting, or is mistaken and forgets to check. It is no longer a link but a chain!

I am still amazed and amused at the same time to see how, over the decades and then centuries, certain great mathematicians, heads full of an algebraic logic that is certainly necessary but that ignores any abstract, general logic, have managed to weave themselves the stitches of the net that enclosed them, cut off from simple common sense. Observing all the traps, obvious to the humble, attentive and unprejudiced observer in which Fermat has placed his best clues, the best conclusion we can draw is: simply sublime. It's difficult to judge these intellectuals, whose thoughts and remarks are both a deep feeling of inferiority towards Fermat and a total ignorance of this man and also of the human soul in general, we can only be amazed by their blindness. When we understands how such a long urban legend could be elaborated, we draw a deep lesson from it, the study of "group think", the observation of rivalries and jealousies between great men, also, reward us in the knowledge of human psychology: how a collective imagination can completely deviate itself.

Three different versions of Arithmetica: first codings/indices

– During more than three centuries, scientists have never read a faithful translation of Fermat's second OBSERVATIO: “I really have exposed a marvelous demonstration of this” (and not “I have discovered a truly marvelous proof of this”). The widespread translation of detexi by “I have discovered” is wrong: if Fermat had meant “I have discovered” (or “found”), he would have written inveni. Latin, the language of scientists and scholars, is a subtle and delicate language to handle. Detexi (from detego), can also be translated by I have laid bare, I really have exposed, or I removed what covered.
– There are 3 different versions of Arithmetica from 1670, where the famous note stating Fermat's great theorem takes three different aspects. Thanks to Mr. Franquart, who told me, in 2009, about his research on the "Observation" present on the Arithmetica at Lyon Library, my passion for this enigma (which the treatment seemed to me shocking) was further increased.

– Version (A), Zurich Library

Once on the site, download the pdf file, then go to page 141.

The word is correctly written, only the final point is overloaded, as on the other editions. Never have we seen an “entire book [devoted to the science of numbers]” the prologue of which (by Diophantus) is longer than the book itself (Fermat), 340 pages against a fortnight. The proof really “exposed” by Pierre de Fermat in the seventeenth century, if it is very short, is extremely difficult. Decryption by R.F. shows that Fermat elegantly used the properties of Pascal’s arithmetic triangle, known since the tenth century. The codings in the Latin text, before being broken, cover, hide, conceal (latin verb: tego, is, ere, texi) his explanation in a masterful feat.

Version (B), Lyon Library

(Roland Franquart): The overload on the t in detexi suggests that this letter could have a big importance for the continuation. Moreover I think, like R. Franquart, that this t has a connection with the last two words of Fermat's observation, non caperet (had it not contained this t, in Pascal's triangle - the full explanation is on Franquart's site). The overload also has the great advantage of forcing attention to the word detexi: "I have unveiled" (or laid bare). The full stop following the word is again overloaded, as if to remind us of the importance of the word that precedes it. In addition, the t initiates texi, meaning "I have hidden". Roland Franquart's deciphering reveals a second reading: "[...] what I have entirely constructed as a fabric is the surprising explanation. The lack (the smallness) of the margin (or border, or edge, limit, frame) would not contain it." These codings and decodings may seem far-fetched to the layperson, but let us remember that Fermat loved to play with his correspondents, and with words (let's not even talk about numbers). He tried to leave as many clues as possible by scattering them all over the place (see below).

The crude overprint on the t appears on several of the copies of the Arithmetica that we have found and these overprints are identical. The printer therefore had to make a new movable typeface specially. It should also be noted that if this "t" had initially suffered from a lack of ink and had not been perfectly visible, it would have been made clearly legible without such a crude overprint.

In June 2017, I spent long hours looking for a quirk that might have appeared in another version of the 1670 edition, preferably on the word (detexi) where Roland Franquart had already found (among other things) the quirk of the overloaded t. I thought that if Fermat had wanted to give his followers the best chance of finding his explanation, he would have risked nothing by using this stratagem a second time. But honestly, I didn't think I could find a third, different version, let alone a second oddity on the same word, it would have been too good. If I was so persistent, it was because I had a crazy hope: to find a "killer argument". And finally I found it, this second big oddity, on the copy of the University of Rome (detexṡ). I could not believe my eyes, this discovery was so unexpected that it left me dumbfounded, and the blow of the sledgehammer was mine. For a long time I remained in this state, not knowing what to think. Personally, I was too involved and it was difficult for me to think calmly about the new situation. This extra oddity "seemed too much", it was "too big", even from the very facetious Pierre de Fermat. But I hadn't considered enough that he was working in an age without internet. It took me almost two years to find the blindingly clear solution. Once on the site, type in the page number 141 at the top, then enlarge the image (+ sign at the bottom right).

– Version (C), Rome University

Once on the site, download the pdf file, then go to page 141.

We observe that the element preceding the end point, strangely, is neither a pure i, nor a pure s, but this strange character, ṡ, which is incongruous in this text. The letter "s" topped with a of a point in chief is a grapheme of the extended Latin, formerly used in Irish alphabet. A diacritic is often used to distinguish a word from another word, a namesake. Why did Fermat, a philologist, transform the word detexi (I laid bare) into detexṡ? This word being unknown in Latin, let us examine the last character, ṡ. It is formed with a i which is twice (crooked), included in the ṡ; the two characters i and s are merged, the grapheme can be decomposed into i and s, which gives us → is. The unknown word detexṡ becomes the word detexis, (verb: detexo, this time, and no longer detego. But detexo = to weave completely; conjugated: 2^nd person, singular, present tense, indicative mode → “you weave completely” (or “you represent completely”), that joins and confirms the alphanumeric decryption performed by R. Franquart in 2009: « I have completely weaved ».
— detexis, from the verb detexo :

Meaning of the verb, on Dictionnaire Gaffiot : to weave completely, to represent completely.
Its conjugation, on Dicolatin (present tense).

Fermat showed great ingenuity here. However the discovery that delighted me the most was not this very curious version of the Arithmetica, because while I spent a lot of time researching a third different version of the Arithmetica, I was very lucky (and stubbornness), it could not have been present on the internet, and finally the decryption was neither too long, nor too difficult, with the data that I already had thanks to Roland Franquart. No, where I was happiest was when I made this discovery about "Fermat's numbers" (see : First link, the bluff: Numbers of the form 2^2ⁿ + 1). When we follow in Fermat's footsteps step by step, we become more and more daring to venture on the other side of the mirror, where no one has gone before. It took time, intellectual availability, meditations, and as a certain aptitude for serendipity so that, after multiple re-readings of the letter to Carcavi over many years, and knowing it by heart, I reread it one last time, as if thinking of something else, and suddenly Fermat's cunning appeared to me in all its splendor. But before that, I must have dared to imagine that his trick could be diabolically clever, and I had to weight each word of the proposal addressed to Carcavi. There is something very pleasing in the fact that we the humble people often have a much more peaceful life, therefore a healthier and finer vision of things, more encompassing also than very prominent personalities subject to all kinds of professional constraints.

On reading a long and warm e-mail that Catherine Goldstein sent me in January 2022, I understood why she could not allow herself to give her opinion on Fermat's proof, since an official recognition of the validity of Fermat's proof would not only cause a stir among thousands of enthusiasts and very time-consuming and embarrassing collateral damage, but above all she did not want in any way to be at the origin of a new polemic that would be harmful for everyone. And to tell you the truth, I ended my reply to her message with these words: "Please accept my eternal gratitude". Without his encouragement during my last few years on Wikipedia, without his messages of support, without the complicity that binds us - a common passion for Pierre de Fermat - I would not have had the necessary motivation to carry out such an in-depth study, which has given me so much joy and so many beautiful emotions, and which I can now deliver to your sagacity and your criticism. It is in a way offered as a "free service", if so-and-so can ignore it, another can be inspired by it and even, why not, make it completely his own. I would be delighted.

I give my promise to you that the following is true. An official of Agence France-Presse was surprised in 2009 that none of the scientific journalists to whom Fermat's proof had been submitted wished to follow up. Are you surprised? The great innovation that Wikiversity has produced (among other things) is that, by using the tools of Wikipedia, it provides independent researchers with the best working support they could ever find, allowing an ever-widening public to access unpublished work that cannot be found elsewhere. If this modest essay could encourage young researchers to understand how sheep-like behaviour thwarts discernment and personal initiative, it would have fully achieved its goal. One feature of this enigma has that surprising and welcome anagram (in french): la légende urbaine rendue inégalable. It seems that logically everything follows in the best way so that this long epic never ends. Could we better thank the great man than by dedicating this study to him? Thank you, Mr Fermat.

Three possible readings of Fermat's observation

« Cuius rei demonstrationem mirabilem sane detexi ».

(Translated from french) ‘’Ludivine Goupillaud wondered about the use of Latin by the mathematician Pierre de Fermat. [...]. According to L. Goupillaud, the merit of Latin, for Fermat, is to be a rigorous language which conforms to the requirements of mathematics, something that vernacular languages do not allow. Long-established language by grammatical norms, it can function easily as a ‘’machine to code and to decode’’, even if, as we can see under the pen of Fermat, it sometimes requires glosses in French to explain the exact meaning terms used. That said, the formality of the Latin sentences, at the same time a pledge of clarity and elegance, allows the fixing of the rules released, without the embarrassment of the explanatory gloss: the brevity – we know how much the mathematicians of the classical age like to skip the intermediate steps of reasoning – provides reaction and activity on the part of the reader, even if it means taking the risk of the enigmatic ellipse or the encryption (don’t we are then in the golden age of the concetto, where the Latin model remains predominant?). If there is sublime in mathematics, Latin is, according to L. Goupillaud, the perfect ‘’marker’’, arousing admiration before abysses opened by mathematical reasoning [...]’’. Emmanuel Bury, ‘’Tous vos gens à latin – Le latin langue vivante, langue savante, langue mondaine (XIV^e-XVII^e siècles)’’, Droz Editions (quotations authorized by the authors and the publisher).

Fermat made sure that his observation could be understood in three different ways:

First reading

Let us recall that detexi can also be translated as ''I have uncovered'', which can easily be confused with ''I have discovered'' or ''I have found'' (which are said inveni, in Latin). We are certain that Fermat had anticipated that the verb he chose (detexi) would easily be misleading, especially in the general context of the observation, which makes his detractors think of a joke, or even a boast. Thus, for the German mathematician Ernst Kummer, Fermat's observation is nothing but a "joke". The mathematician who does not have a perfect command of Latin will translate the text as follows:

I have found a truly admirable demonstration of this. This reading is for the followers of the followers who have only a few notions of Latin. Moreover, they have not studied Fermat's psychology, his predilection also for pedagogy, for the hard work that brings the greatest satisfaction.

Second reading

The confirmed Latinist, especially if he has no unfavourable prejudice towards Fermat, will have studied the Latin text very closely and translated it correctly:

I really have unveiled the surprising demonstration

or: I really have exposed the surprising explanation

Third reading

The Latinist can further deepen the formulation by taking into account this famous detexis (on the 'Rome' edition) of the verb detexo → 'to weave completely'. Roland Franquart had found exactly the same denomination, but thanks to the 'Lyon' edition and a completely different coding. We then have a third reading:

I have actually woven, completely, the surprising explanation.

As we have seen above, by combining these last two readings, we can add a fourth, more complete one:

I have really unveiled, fully woven, the surprising explanation.

One is either a pedagogue or one is not. Note that the words ''text'' and ''fabric'' (and ''embrace'') have the same Latin root, textŭs. Note finally that in the last sentence of the comment, the word marginis can be translated as ''margin'', ''border'' and ''boundary''.

"Concision, in addition to its stylistic virtues, plays a role as a stimulant, especially in epistolary exchanges. By deliberately withholding his conclusions, by revealing only the lineaments of his thought, Fermat creates an emulation through ellipsis [...]." Ludivine Goupillaud, “Tous vos gens à latin.”

The formality of Latin sentences, both a guarantee of clarity and elegance, allows the rules to be fixed without the embarrassment of explanatory glosses: conciseness - we know how much mathematicians of the classical age like to skip the intermediate stages of reasoning - provokes reaction and activity on the part of the reader, even if it means taking the risk of enigmatic ellipsis or encryption (are we not then in the golden age of the concetto, where the Latin model remains predominant? ) [...].<ref>On page 181 of the Varia Opera Mathematica, Fermat interrupts his presentation in French and writes: "I will tell you in Latin because the French language is worthless there."<ref> This is followed by eleven lines relating to the eighth question of Diophantus, from which he derived his famous observation on the great theorem.
Emmanuel Bury. “All your people in Latin.”

We claim that Fermat knew that his sentence, which we would translate in the way that would suit us best, would mislead the "followers of the followers" who would see in him only a boaster or a giddy man (as you wish). It is the same kind of subterfuge that he used when evoking the famous false conjecture in the letter to Carcavi. One is either an artist or one is not.

"Philosophers of science pay particular attention to language: they develop the idea that the experience of Common Sense, expressed in everyday language, should serve as a basis for theoretical scientific discourse: indeed, the truth value of statements in everyday language is superior (in its recognition) to that of statements in scientific language." (Marie-Anne PAVEAU).

Fermat's tricks are remarkable. Thanks to Roland Franquart who discovered the first and most important clues, and who, having learned that I was interested in this enigma, informed me of it in 2009. I take back here the most symbolic ones (noted RF), sometimes by modifying them somewhat (I hope not to betray his thought too much), and I add those found by myself (CM) and other authors.

Common codings to each version

I have assembled here in Fermat's note the 2 oddnesses on the word detexi appearing in 2 different editions.

C^Vbum autem in duos cubos, autem quadratoquadratum in duos quadratoquadratos
& generaliter nullam in infinitum vltra quadratum potestatem in duos eiusdem
nominis fas est diuidere cuius rei demonstrationem mirabilem sane detexṡ_•
Hanc marginis exiguitas non caperet.

Fermat's tricks are remarkable. Thanks to R.F. who discovered the most important clues and informed me in 2009, as he had learned that this enigma fascinated me. I'll repeat here the most symbolic, sometimes modifying them somewhat (I hope not to betray too much his thought), and add those found by myself (CM) and other authors.
– (Roland Franquart:) In the first word of the Observation, C^Vbum (cubum: cubic number), the exponent, like in every first word of a paragraph, on page 61, should have been entirely written in capital letters: C^VBVM. Since in Latin, u written in capital becomes V, the second V is missing. This transgression is one of the transformations which allowed Fermat to perform its encryption. It is also a strong clue, visible in the very first word of the Observation. If in its 47 other Observations Fermat would not have made the same transgression, the anomaly would have been too much blatant (CM), for the edition of Rome in particular (detexṡ). These two oddities on the letters t and u, direct us to the choice of these two letters to attempt a weaving. This begins to be confirmed since in its note we find 21 u (and u is the 21^e letter of alphabet), and only 19 t (t is the 20 ^e). So one t is missing. Roland Franquart explains that this lack is to be put in relation with the last two words of the observation, non caperet: would not contain → this t, the importance of which is further increased by the point which follows the word detexi. The point is also overloaded in the two other versions. These “coincidences” between the number of letters u and t, and their respective rank in alphabet strongly incite us to choose them as elementary bricks of an coding.

– (R.F.) In the note there are 2 couples of letters ut in order, and 3 couples tu in order. Since Fermat says to us "you weave completely", Roland Franquart made a weaving, as simple as possible, with the letters ‘’u’’ et ‘’t ‘’, that he introduces in Pascal's triangle. All this seems quite complex, but it seems that Fermat had no choice if he wanted to code his explanation in three 1/2 lines. With this coding he was also lucky, the couple "tu" is also the personal pronoun "tu" ("you") which is placed in French before "tisses" (weave) and which increases the importance of the "tu" already present in the literal translation from Latin detexis into French. He spells two others words, in his observation, in a personal way (vltra and diuidere instead of ultra and dividere).

– (CM) In the wording of his OBSERVATIO the repeated presence of letters u and t in the declensions and variants of the word quadratus (square number) certainly guided his choice in the use of these two letters to organize his text in such a way that the codings give the impression of a deliberate will. It's amusing to note that the expression quadratoquadratum in duos quadratoquadratos (square of a square in two squares of squares) refers to the case n = 4, the proof of which appears in the only theorem that Fermat fully explained in his 48 observations.

Across the centuries, many scholars have doubted that Fermat really had a proof. After Andrew Wiles's discovery in 1994 – a proof of tremendous complexity – they were even less able to imagine it, having taken more than three centuries to prove it. Some others, more objective, officially say that they do not know what to think. That is the case for Jacques Roubaud, Catherine Goldstein, a world-renowned expert in Fermat'works.

Fermat's codings discovered by Roland Franquart are so obvious that we think: "They can't be coincidences, it's just a masterful exploit". In the only wording of his observation we already find 9 curiosities. After a new decoding we find 4 others, literally amazing. Then in his correspondence we find new ones. Let us quote Fermat in connection with another of his theorems: "I cannot give the proof here, which depends on numerous and abstruse mysteries of the Science of numbers; I intend to devote an entire Book to this subject and thus make this part of Arithmetic astonishing progress beyond the limits formerly known." D.P. F. n° XVIII). Like with all of his other "theorems" (except one) which were all later demonstrated, he does not deliver his proof to Digby. It will be necessary to wait 175 years to have a proof by Cauchy (1813).

We know the role of the psychoanalyst, who doesn't have to reveal to the person (rightly called the analysand) lying on the couch, some of the unconscious thoughts that he-she might have discovered in the person over the course of the sessions. Nor does he have to reveal to this analysant the mechanisms at play. On the contrary, he lets the analysand say everything that comes into his-her head at instant t. From time to time he would say a few words to open a lead, give a clue, but he would never reveal what is unconscious for the moment, to which the conscious mind, thanks to a protective and necessary filter, does not yet have access. The psychoanalyst is above all a psychologist, an honnête homme, intelligent, empathetic, having already done a profound work on himself. Fermat was not a psychoanalyst, he was above all a great mathematician, intrepid, and above all the honnête homme par excellence. He had no patients, only correspondents who were not at all patient. Very few of his correspondents (Mersenne, Pascal) could understand him. He acted with the mathematicians of his time and those who would follow them in the manner of a persevering and sagacious psychoanalyst, who would have dealt with cohorts of patients who came here without even really believing in psychoanalysis. Knowing their lack of confidence, without helping them in their work, he had to provide them with as many clues as possible, hoping that one day one of these mathematicians would finally come out of his apathy, lie down on the couch and hear in his mind a few key words.

Already in 1637 when Fermat sends Marin Mersenne his method of researching maxima and minima, he doesn't take the time to set out the arguments he used. Maryvonne Spiesser, mathematician, historian, honorary university lecturer, explains about this method:

"You have to read Fermat's text "forgetting" our current mathematics, especially the idea of limits [...]."

I completely agree with the idea of this example. Better still, this conception is for me the only way to access the understanding of the proof he gives. In order to understand Fermat, one must already WANT to understand. He who doesn't even try, obviously never will. Moreover, was it easy for a professional mathematician used to reading calculations and demonstrations to imagine, even at the sight of a strange anomaly in the "Observation", that it would be necessary to look for other clues that Fermat might have left? With a lot of luck this could have been done in the first decades after 1670.

Around 1800 the great theorem could be proved for values of n equal to 3, 4 and their respective multiples, then, with a first great advance due to the work of Sophie Germain, for n=5, 14, 7. Fifty years later, when mathematicians were desperate to find an arithmetic proof of Fermat's last theorem still to be proved, Ernst Kummer began a bend that would give a completely different turn to the case. Radically changing the approach he had the idea of using complex numbers, developing the theory of ideal complex numbers, which was to become a very important tool in algebra. Finally he demonstrates the theorem for all exponents below 100 and takes this opportunity to talk about Fermat's theorem as a simple curiosity. But it is a new advance which, even if only very partial, arouses enthusiasm among the scientists who until then had made little progress. But the trick was taken and pure arithmetic research was definitively abandoned in order to try to demonstrate the theorem, all the more so as the new path is full of promises for new mathematics. From now on they will devote themselves to exploring this new, strange and complex species of numbers, these ideal complex numbers, which will help to go much further in the understanding of prime numbers, by studying the deepest mathematical questions. Jacques Roubaud notes that from this moment on, it becomes impossible for a mathematician who does not possess as much knowledge in arithmetic as Fermat does, to have access to his reasoning. They will therefore start studying Fermat again, but differently. Yes it will be difficult, yes it will be complex, but at least hope has returned, and above all they doubt even more that Fermat could have demonstrated his theorem. Then, over the centuries, as scientists use Latin less and less, and mathematicians demonstrate the theorem for particular cases (and even more so after Kummer "invented" the theory of ideal complex numbers - taking the opportunity to talk about Fermat's theorem as a curiosity, a joke), no one thinks to look closely at the original observation. It therefore seems logical that it was an amateur (Roland Franquart), who went directly to the source to closely examine Fermat's note written in Latin and to highlight Fermat's codings.

Eventually, commentators missed little, just a little cunning, humility and above all confidence. Knowing however his facetious spirit, they did not question in any way the reason that Pierre de Fermat, a Frenchman - who nevertheless addresses himself, in the first place, to French people - may have had to write the observation he considers (it is difficult to doubt it) as of the utmost importance, to write it in Latin, and only in Latin. It is true that here again he muddies the waters: he does the same with the 47 others. When you think about it a little, going first to see what the only true source, the Latin note, says, seemed the most rational, the most relevant approach. Nor did they have the idea of turning to a professional Latinist in order to get the most accurate translation possible. Nor did these commentators think to rely on the official translation of Émile Brassinne, almost perfect, published in a mathematics book. It's true that this translation was relatively late (1853), long after the publication of Arithmetica. And already... Kummer had been there, developing his theory. All his colleagues rushed into the breach and stopped studying Fermat by the resources of pure arithmetic. Because, surely, Fermat “is boasting”, or “he was mistaken”, or “he understood that he made a mistake but did not see fit to retract it, unless he is mocking us, and anyway he did not have the right tools since he only had his own ones!”

Since the Arithmetica of 1670 has been published, one cannot doubt that mathematicians (French, English, German...) have read the Observation in one of the two "arranged" versions (detexṡ. or detexi.). Is it easy for a professional mathematician, used to reading calculations and demonstrations, to imagine, even at the sight of a strange anomaly, that it is necessary to look for other clues that Fermat could have left? With a lot of luck this could have been done in the first decades that followed.

For those who had read the observation on one of the two editions of the Arithmetica of 1670 altered by Fermat, this would have been possible. When Fermat writes that he really has unveiled an astonishing (admirable) demonstration, they might have thought that this demonstration was very unusual. Although the presence of codings is obvious, Fermat's explanation is far from being easily accessible to twenty-first century mathematicians - when they want to think about it without preconceived ideas. Moreover, after more than 350 years of wanderings, is it easy to admit that all the arguments put forward by Fermat's detractors during the long history of this theorem are specious? By herd behaviour, over the decades, commentators have taken these arguments and expanded them, building a huge and completely flawed thing. Conforming to the dominant thought is comfortable, which avoids speaking out and feeling apart from one's caste. What courage it would take to face the pretenders to the infused science, and the cohorts of scoffers and nitpickers: "If there was an ounce of truth in all this, for a theorem so important to Fermat, he would have put all the details. "To make our work easier? In the margin? Between the lines?

The observations that Samuel Fermat inserted in the 1670 edition are written in an impeccable style and the two oddities on the same word in 2 of the 3 versions are obviously voluntary. Historians of mathematics are first and foremost mathematicians and base themselves above all on explicitly reported calculations, and generally on precise facts. Moreover, they are very rarely Latinists, especially nowadays. In 1995, in her book, Catherine Goldstein was more refined: "In any case, this approach [by Andrew Wiles], in which Fermat's theorem is only a very attractive but minor corollary, is based on recent Galois representation techniques. It remains possible that a direct elementary proof can be found. "(page 120 of the book, note 7). Through its technological progress and lack of faith, Humanity has become increasingly proud, believing itself to be self-sufficient. The most perverse corollary of this pride is pessimism (individual, societal) which in turn reinforces pride. This pessimism leads us away from the simplest ideas, the only really effective ones.

Roland Franquart website: franquart.fr

Pascal's triangle

“Intellectuals solve problems, geniuses avoid them.” (Albert Einstein)

Pierre Fermat was quite the opposite of a sheep. It is not surprising that he was such a passionate searcher: far from Paris, remote, he had no contact with the other mathematicians except by letters: he was justified, in his intellectual solitude, to appreciate the most arduous and rewarding researches. Fewer and fewer people were able to meet his challenges, and finally no more could do. Huygens was interested in other things; Pascal, for his part, had moved away from mathematics and entirely entered in theology. On August 10, 1660, Fermat, as his health was declining, urged Pascal, much more sick than him, to accept an encounter to “converse with you for few days”. He wrote: “Our thoughts adjust so exactly that they seem they have taken the same route and follow the same path: your last treatises on the arithmetic triangle and on its application are is an authentic proof of this [...].” Was it because both of them were ill that he would have liked, by telling him his discovery thanks to the arithmetic triangle, that both could share a moral comfort each other? Fermat could have knowledge in arithmetic triangle even before Pascal, thanks to the works, that he knew, of François Viète, who died five years before his birth (this triangle was already known in the eleventh century by the Persian mathematician Al-Karaji, and by many others following him, until Tartaglia, Viète...). So if Fermat was interested in the astonishing properties of the arithmetic triangle — we remember that he worked on magic squares, and to play with this philosophical object that are numbers, to study all relations they can have each other, represented a treasure to exploit —, it seems logical that he never wished to refer to anyone until Pascal himself speaks about it, if he used it for his great theorem. This step aside that he would have done to circumvent the obstacle, using the arithmetic triangle, could it have allowed him to find a proof? The decoding of the observation that Roland Franquart made in 2009 reveals in broad outline what kind of proof it would be. The fact that Fermat fought so hard in coding his note shows that he was certain of the correctness of his proof. If the decoding is right, it seems that Fermat has provided a very thorough work about the axioms, which leave us puzzled.

M.P.E.A.S.

Pierre de Fermat never signed any work using his name, it was using a pseudonym that his only published treatise (on geometry) was published in 1660. On the cover of the book, following the title, the author's signature looks like this:
Autore M. P. E. A. S. (following a librarian's addition: ‘’de ferm’’ (i.e. ‘’de Fermat’’)). Then under the image, the word TOLOSÆ.
Here is what was said in 2001 (page viii), in the book 17 Lectures on Fermat Numbers-From Numbers Theory to Geometry (Canadian Mathematical Society, Springer Editions):
“Indeed, he published only one important manuscript during his lifetime, and signed it using the cryptic initials: M. P. E. A. S. Their meaning remains inexplicably unknown."
Roland Franquart's explanation (2014):

Magistro Procuratore Enodare Apud Sedem (TOLOSÆ) (Magistrate Procurator Investigator At the Seat (TOULOUSE).

Review of the research

- We have seen that none of the three arguments put forward by Fermat's detractors are valid. They can also be reversed in favor of Fermat.
- Let's gather all the favorable arguments:

1) The style used by Fermat in his 48 observations (written where?) clearly shows that they were written for the reader.
2) Samuel did not keep the copy of the Arithmetica where his father's 48 observations were supposed to have been written, even though it would have had considerable historical value: it simply "disappeared"

Regarding the observation about the great theorem:

3) It is the only one of the 48 whose title is not abbreviated in D.P.F. but written in full.
4) C^Vbum . The exponent is not written according to the usual rule.
5) There is a first version with a ‘’detexi’’ that is différent (‘’detexi’’).
6) And a second version with a different ‘’detexi’’ différent (‘’detexs’’); and a third version without any anomaly on this word.
7) detexi does not mean I have discovered (or found) but I have unveiled (or laid bare, exposed).
8) ‘’detexṡ (⇒ detexis), on the edition of Rome, is translated exactly by ''you weave completely''. The expression ''I have woven all of it'' had already been found thanks to another (more complex) coding discovered by Roland Franquart on the edition of Lyon. The two occurrences reinforce each other.
9) The adverb sane (really), by the unusual way it is placed, applies to both detexi (surely unveiled)...
10)... and to mirabilem: (really) astonishing, or admirable, or wonderful.
11) The point that follows the word detexi is enlarged (different from the final point) in the three editions to further emphasize the word detexi.
12) Ludivine Goupillaud, former researcher and now teacher, noted before us that Pierre de Fermat takes "the risk of enigmatic ellipsis or encryption".
13) La répétition dans l’observation, d'abord des couples de lettres ‘’tu’’ (3 fois), puis ‘’ut’’ (2 fois).
14) We find 21 u (and u is the 21st letter of the alphabet), and only 19 t (but t is the 20th, so a “t” is missing, the cause being here : "non caperet" (would not contain this t, t which has been highlighted in detexi and comes into account in the exploitation of the arithmetic triangle (the detailed explanation is on R. Franquart's website).
15) These singularities allowed R. Franquart to "weave completely", like Fermat did, the explanation, the demonstration.
16) On August 29, 1654, Fermat wrote to Pascal: “Our thoughts fit so exactly [...] your last treatises on the Arithmetic Triangle and its application are an authentic proof [...].” Now, the thesis developed by Roland Franquart and myself after him is precisely centered on the arithmetic triangle.
17) Fermat interchanged the letters v and u in the words “vltra” and “diuidere”, but these two words do not exist, only until the 15^th century was u written as v - but the reverse, v → u, was not possible. This inversion reinforces again the importance of an argument exposed in Part III: first, to give great importance to these two letters in the encryption of the whole note, then even if one easily understands that the pseudo-word "diuidere" corresponds to the word dividere — and here I copy the words of R. Fanquart: "Fermat shows to us that the letters “u” and “v”can replace each other in any word of the ARITHMETICA, except in the first word, the ^EXPONENT.
18) From his complete proof of Fermat's right triangle theorem (Obs. XLV) one immediately deduces the proof of the particular case n=4, but Fermat passes over it in silence. We wrote in conclusion what we had to understand: he knows what he is talking about and, very discreetly, if one wants to read between the lines, he lets us know.

Regarding Fermat numbers:

19)The letter to Carcavi where the false conjecture of Fermat numbers appears, that some people have interpreted in an obviously oriented and irrelevant way, his son does not take it up in the Varia.
20)This conjecture is also absent from the observations of the Arithmetica of 1670, all of which were later proven correct.
21) Fermat writes in his letter "I have considered", not "I have demonstrated that [...]".
22) Fermat uses the expression negative questions, which he never does elsewhere, the established expression being negative propositions. Which in the context easily suggests a double meaning: "the answer to this question is negative".
23) The arrangement of singular formulations throughout the paragraph allows for a second reading: we saw earlier that the phrase "This last question is of a very subtle and ingenious research [...]" (with two duplicating synonymous adjectives), suggests that the formulation of this "question", in its context and with such a peculiar formulation, must be for us the object of a very subtle and ingenious research.
24) Mathematicians agree that Fermat knew the method (with divisors of the form 64k+1) showing that these "Fermat numbers" are not prime.
25) We have seen that the letter to Mersenne of June (?) 1640 where Fermat uses a similar method (divisors of the form 74k+1), his son also omits it from the Varia.
26) Fermat evokes the famous false conjecture... 6 times over a period of... 19 years... to all his correspondents... and asks for their help (...) to try to prove... that it is true (...).
27) Letter to Frenicle de Bessy (October 18, 1640): "[...] for in advance I warn you that, as I am not capable of attributing to myself more than I know, I say with the same frankness what I do not know [...]". This letter, the only one about this conjecture that Samuel chooses to insert in the Varia, is the one in which his father says he is always honest. It seems to indicate that it can be related to the observation about the great theorem (i.e. we should take this observation seriously).
28) Fermat's observations could not have fit in a margin, some of them are far too long.

We count 28 arguments in favor of the existence of a proof by Fermat. Two others can be found in the chapter M.P.E.A.S.; others are more related to pure psychology, we mentioned them in this study; others again, more complex (by alphanumeric decryption) have been developed by Roland Franquart on his site.
Fermat, by not publicly revealing the proof of his most famous and most difficult theorem during his lifetime, made it the high point of his research (Andrew Wiles had to work on it in secret for 7 years and write 1000 pages). And to confuse the issue, he wrote many of the other 47 observations in the same facetious and provocative way (lack of space or lack of time), with the same terms "admirable", "really difficult", "very beautiful", "general method", "infinity", etc. Thus historians will not have the idea of looking for anything unconventional in the famous observation (anomalies and "coincidences" too extraordinary not to be voluntary). If one hardly pays attention (or not at all, often) to the note written in Latin, as all historians except Tannery have done for 350 years, without having a global vision of the Fermat affair, one does not pay any attention to the title of the OBSERVATIO (not written in abbreviated form as elsewhere, but in all letters, CAPITALS moreover) which explicitly asks us to OBSERVE.

As for the argument that the 48 observations were not written for the reader, it is absurd. To claim that Fermat would have said that numbers of the form 2²ⁿ + 1 would all be prime is equally absurd.
All these people have not only limited, but above all perverted their imagination by disregarding the purest logic, and with a lot of bad arguments in a single desire, a quasi-existential need: to denigrate Pierre de Fermat - to claim to be more subtle than this immense genius, what a pleasure!
No thought can capture the sublimity of Fermat's prowess as it unfolds in the darkness of academic unconsciousness. The silliness revealed by the common thought that has accompanied the whole epic of Fermat's Last Theorem has made it the most ubiquitous urban legend, the most incredible too.

I am so familiar with the academic mindset that I am convinced we will never get the official end of the story - which gives it a delectable charm. The suspense will continue and we mathematicians and puzzle lovers can continue to meditate on human nature, its quirks and weaknesses, to our advantage.

"It is easier to break an atom than to break a prejudice."A. E.

Fermat and the publication

If he made known by mail some of his short manuscript treatises, most of them devoted to geometry, he never published anything under his name. Perfectly aware of his worth, he knows how to assert himself, but in the opinion of those who know him well he is very modest. As we have seen before, his epitaph ends with these words: “Will you not ignore what is useful? Know how to be ignored ". In 1659 he “apparently” tried to have his work published by soliciting the active contribution of Carcavi and Pascal, who were responsible for putting everything in order in his writings and finding a publisher. He tells them that the book should not bear his name:

August 9, 1654 most certainly (1659 according to another source).

Letter from Mr. FERMAT
TO M. DE CARCAVI

Sir,

"I was delighted to have had feelings in line with those of Mr. Pascal; because I infinitely esteem his genius and I believe him very capable of coming to the end of all that he will undertake. The friendship he offers me is so dear and so considerable to me, that I think I should have no difficulty in making some use of it in the printing of my Treatises. If it did not shock you, you could both produce this impression, of which I consent that you are the masters; you could clarify, or increase, what seems too concise, and relieve me of a care that my occupations prevent me from taking. I even want this work to appear without my name, giving you, apart from that, the choice of all the designations that can mark the name of the author, whom you will call your friend. Here is the bias I imagined for the second part, which will contain my inventions for numbers. It is a work which is still only an idea, and which I would not have the leisure to lay down at length on paper, but I will succinctly send to M. Pascal all my principles and my first demonstrations, of which I answer you in advance that he will draw things not only new & hitherto unknown, but also surprising. If you join your work with his, everything can succeed and be completed in a short time, and yet the first part, which you have in your power, can be brought to light. If Mr. Pascal tastes my openness, which is mainly based on the great esteem I have for his genius, his knowledge & his mind, I will first begin to tell you about my digital inventions. Farewell, I am, Sir, your…"

This cavalier letter asks, did he really think that Pascal and Carcavi would agree to get down to shaping all his discoveries, which would have required a lot of work and energy from them? Or did he never intend to publish all his demonstrations and “reveal all his secrets”? In his eyes, his discoveries were not appreciated at their fair value and the “important book” that he said he wanted to devote to arithmetic was never published, at least in the form that the public would have liked. Could this letter be just a convenient alibi? → (“I will often indicate, in my Diophantus, that I have, either not the time, or not the place, to deliver the proofs of my theorems. It is more fundamental to say that I have none no desire, but by this letter one will believe that I had wanted to “chew the work” of my detractors – which would have been completely unworthy of a pedagogue!»). His contribution to the theory of numbers will be known by his correspondence and especially by his 48 observations where he will have put all his application, and which Samuel, charged by his father to ensure the publication after his death (this is our thesis) will insert in the Arithmetica of Diophantus.

A brief history of Wiles' discovery

(I put here a great part of what I had written on fr.wikipedia).
The theorem is finally demonstrated by the mathematician Andrew Wiles, after eight years of intense research, seven of which in total secrecy. The demonstration, published in 1995, uses very powerful tools from number theory: Wiles proved a particular case of the Shimura-Taniyama-Weil conjecture, that we know for some time already, via the works of Yves Hellegouarch's in 1971 (note to CRAS), then from Gerhard Frey, Jean-Pierre Serre and Ken Ribet, that it implied the theorem. The demonstration calls upon modular forms, Galois representations, Galois cohomology, automorphic representations, a formula of traces...

The presentation of Andrew Wiles' proof has been made in two stages. In June 1993, at the conclusion of a three-day conference, he announced that Fermat's great theorem is a corollary of his main results presented. During the months that follow, the final version of its proof is submitted to a team of six experts (three are usually enough) appointed by Barry Mazur; this step is commonly called peer review. Everyone has to evaluate some of Wiles' works. The group consists of Nick Katz and Luc Illusie, whom Katz called in July to help him. The part of the proof for which he is responsible is in fact very complex, they first try to apply the Euler system. Among the experts are also Gerd Faltings, Ken Ribet, Richard Taylor and Peter Sarnak, a close friend of Wiles, whom he had taken into his confidence before the June conference.

If Wiles finally succeeded in proving the Taniyama-Shimura conjecture, the end goal, the Fermat, would be achieved, and the repercussions would be considerable in number theory. The nervous tension is very palpable during all these checks as the wait outside is great. All must work in the greatest confidentiality, the weight of secrecy is heavy to bear. After Nick Katz has given to Wiles a few points for clarification, which will be quickly clarified, things start to take a turn for the worse and Nick Katz and Luc Illusie end up admitting that it cannot be established in the evidence, only to apply it, the Euler system, while this element is considered vital.

Peter Sarnak then advises to get help from Richard Taylor, a former student of Andrew. However, attempts to close the void are proving increasingly desperate. Andrew, who until then had worked alone and in secret, is now in the spotlight, It's difficult for him to bear, and after all these efforts, completely exhausted, he thinks he has failed and resigns himself. Nine months later, in autumn, a decisive event occurs. Richard Taylor suggests returning to the attack line (Flach-Kolyvagin) used three years earlier. Andrew, although convinced it wouldn't work, agrees, but mostly to convince Taylor that it couldn't work. Andrew works for about two weeks and suddenly (September 19, 1994):

"In a flash, I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant."

Taken separately, Flach-Kolyvagin and Iwasawa were inadequate, together they complement each other. On October 25, 1994, two manuscripts were published: "Elliptical modular curves and Fermat's last theorem", by Andrew Wiles, and "Ring theoretic properties of certain Hecke algebras" , by Richard Taylor and Andrew Wiles). The first, very long, announces among other things the proof, while relying on the second for a crucial point. The final document was published in 1995.

After that, John Coates, who had supervised the theses of Andrew Wiles and Catherine Goldstein, said:

"I myself was very skeptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime."

Wiles and Fermat

Andrew Wiles said: "I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof." If I had been in his position, I would certainly have answered the same thing (maybe even exactly), it would have been very comfortable for me, these seven years of sustained efforts would not have been in vain (even if this work has greatly enriched the math, but that's another question). Wiles is a great mathematician, just like Fermat. It is pleasant to note that the magistrate Pierre de Fermat, who could not waste the little time he had available to detail all his calculations (almost never keeping a copy of a work transmitted to a correspondent), obstinate as he was to always go further, called himself "the laziest man in the world". Both of them, each in their own way, with the tools of their time, have made considerable advances in mathematics. These two geniuses are a bit like twins. However, Andrew had a handicap, he had to assimilate a great deal of twentieth century mathematics in his training, and had to invent many new ones. Perhaps, if he had lived in Fermat's time, he would have had to be satisfied with more element-ary, purer, more powerful mathematics, which trie to understand the deep relationships between numbers as closely as possible, he would have been able to approach the master. The Ancients did not yet have their minds cluttered with this multitude of complex data that the Moderns were forced to assimilate in order to perpetuate technological progress. Wiles was so amazed by his success after all his efforts (and a big problem towards the end, uncovered through peer review, which seemed insurmountable and was finally solved), that any thought of the existence of seventeenth century evidence could only stumble on the contours of his mind, wonderfully fulfilled by his discovery, nothing could alter his joy. One can try to imagine what it must have been like, when he searches for the words to express it the emotion is so strong that tears come to his eyes. The race for the "Last Theorem" was a long quest of 324 years. Its history is so exciting for mathematicians who have never cracked Fermat's secret that the urban legend that has coexisted since the beginning to ward off an irritating and unbearable spite will logically have to go on quietly. What a fascinating subject, combining sociology, philosophy, psychology, historiography, what an admirable lesson in pedagogy too.

The urban legend

Some commentators have claimed, often with unthinking arguments, that Fermat could not have found a proof to his great theorem. Any serious mathematician familiar with Fermat's work might say that none of these arguments stand up: all of these people have fallen into magical thinking. This rumor, which was comforting, has spread and grown over the centuries, sometimes by naivety, sometimes by conflict of interest, sometimes both of them, always adding mystery to mystery. If Fermat, obsessed with his desire of generality, never mentioned elsewhere than in his observation the general theorem, we know he had always had it in mind. He writes that he does have a proof, yet he never speaks about it during his lifetime, preferring that the theorem would be known only after he dies. In this affaire that looks like a thriller he shows prodigious mastery and virtuosity, both blurring tracks and leaving many clues. He reveals to his only followers the beginning of an explanation with three lines and a half of Latin — though he (Pierre + Samuel) — wrote them differently (hardly) in three versions of 1670's edition; this is something like sublime art. The only edition available in Zurich would certainly not have been sufficient for a decoding, the one in Lyon would have been, the one in Rome, the most revealing (the hidden word “detexis” → “you weave completely”) and eccentric version was of about equal strength. The anomalies / indices in the last two ones reinforce each other, and even more when they are added to the five other ones, and still more when they are grouped with those present in his correspondence.

In his testament-letter to Carcavi for Huygens, where he still makes no allusion to the great theorem, he concludes with these words: “And perhaps posterity will be grateful to me for having made known to her that the Ancients did not know everything, and this relation will be able to pass in the spirit of those who will come after me for traditio lampadis ad filios, as the great Chancellor of England says, according to the sentiment and the motto of which I will add, multi pertransibunt and augebitur sciencia(*).
(*) “they will be numerous to go beyond, and scientific knowledge will be increased.”

Christophe Breuil, a mathematician, gives us some reflections that help to understand the psychology of the scientist.
“Here is for example another little story (another joke) I heard from another colleague who is not so young (but not less brilliant). To find out if the new result that we have just obtained is interesting, it must be done as follows:
1) Modestly explain to a great expert on the subject.
2) Analyze his reaction: if he is happy, the result probably has little interest, but if he scowls, then all hope is allowed. It seems seem to be the "destiny" of mathematicians: that of tackling superhuman problems that cause indifference and misunderstanding of the outside world. But there are the maths themselves, their objects and structures of infinite richness, their beautiful and powerful concepts, their deep unity, perpetual source of renewal and rejuvenation!”
“Any researcher will tell you that emotional or egocentric considerations (and more generally ‘‘human’’ considerations) invariably come and disturb the limpid course of logical reasoning, or fog a mathematical intuition that is taking shape.”[3].
The explanation found by Roland Franquart is of little interest to Fermat's commentators, who are disconcerted and opposed to the conventional way that Fermat had to expose it: he seems to have left aside Euclid's axioms (the method of infinite descent being inoperative), and that makes us puzzled. I do not know personally what to infer. The minimum we could do was to pay tribute to man's pedagogy. Meditating on this enigma, and especially on its history, is very instructive for anyone in search of historical truth, or interested in urban legends.

His most famous Observatio, could Fermat be assured that an extremely hermetic explanation, which he hid there, would one day be discovered? Certainly not. Anyone who could have followed him in his research had definitely relinquished. What does a teacher do when all of his students, one after the other, left the class? What does a scientist do when nobody wants to follow him anymore, when age comes and health declines? What resource is left to a pedagogue who has always longed for progress in knowledge?

If one looks for the whole book that Fermat would have devoted to the science of numbers, there is only to read the Arithmetica of 1670 which includes 48 stimulating observations. Did they help mathematicians to push away the boundaries of the science of numbers beyond the old known limits? The nugget which is there is a tall story that leaves us stunned. Never we saw before, never we'll see again, a universal genius giving the explanation of a powerful theorem in the form of a huge joke, which makes us pensive: "I really have exposed the quite surprising explanation there is no room in this margin".

"Genius cannot be imitated, it cannot be communicated. It cannot be transmitted because genius itself would be incapable of giving the rules, it is on the side of the sublime rather than the beautiful." Hélène Frappat.

Epilogue

What attitude could have a contemporary scientist towards Fermat's clues? His conjecture took up the simple and famous Pythagorean theorem, and multiplied the exponent 2 by infinity, which made this new theorem absolutely fascinating. For more than three centuries scholars have been working on the problem without ever coming close to a general proof – though proving the conjecture for half of the whole n prime numbers. It has sometimes been thought that this conjecture could not be proven, while thousands of enthusiasts around the world were persuaded to have found a very simple proof. Over time, mathematicians have lost interest, especially since they didn't see the usefulness of proving it. In 1993 a totally unexpected event occurred, it seemed that British mathematician Andrew Wiles was not far to solve the problem, the event was reported on the front page of whole world's newspapers. Eventually, in 1995, he completely succeeded. Since then, mathematicians, finally rewarded for all their efforts, have become more and more disinterested in looking for a more ‘elementary’ proof, which should then be very difficult. They had doubted for centuries, it had been frustrating, nevertheless they were resigned. But in 1995, when Wiles, helped by Richard Taylor, published his proof, the excitement is commensurate with the discovery, all the frustration that had been accumulated instantly swept away.

Our mathematicians no longer want to renew an experience that has already taken them too long. The same who claimed that Fermat must have been wrong may still affirm it. The minimum we could do was to salute the pedagogy of Pierre de Fermat . Meditating on this enigma, on its history, on its actors (who may be interrogative or most of the time peremptory), is constructive for any researcher in search of historical truth, or simply interested by urban legends.

His most famous observation, could Fermat be sure that a demonstration that he would have hidden, highly hermetic, could be discovered? Certainly not. Anyone who could have followed him in his research had definitely forsaken him. What does a teacher do when all of his students, one after the other, have left the class? What does a scientist do, as age comes and health declines, if every one no longer wants to follow him? What resource is left to a pedagogue who, always, had ardently wished for progress in knowledge? His approach having always been mutual stimulation, he keeps the same approach. For those who might, one of these fine days, accept yo to take up the torch, he delivers, without facilitating their work, he delivers 48 brief and valuable observations. Sometimes he does not have room, sometimes he does not have time to expose a demonstration (always admirable) of what he advances. He only once gave us the complete proof of a theorem. My certainty is that mathematicians, occupied on their own work, are not at all interested in the clues left by Fermat, and that they have definitively closed an already too long history. Fate made that Fermat and Pascal could not meet in 1660, this same fate now seems to suggest that Fermat's enigma will never be resolved, and that gives it, anew, the attraction it had lost.

All of Fermat's correspondents had forsaken him one after the other despite having a good idea of his abilities, then should not he feel some bitterness that his contribution to the science of numbers and his merits were not considered at fair value? Let us not underestimate Fermat, minimize his discernment. He was aware that one could take him for a boastful (he had played with this) with his unorthodox and provocative ways. The meeting with Pascal could not be done but the letter remained to us. Did Fermat think that his followers might be interested? Anyway he was definitely lonely. Quite disappointed, he consoles himself by mounting a challenge to the world.

The moral of a never ending story

.../... To do.

Amazing anagrams

(French): Pierre de Fermat préféra méditer. Petri de Fermat permettra défi dernier théorème, étreindre Homère.

English meaning: Pierre de Fermat preferred to meditate. Petri de Fermat will allow challenge last theorem, embrace Homer.

(French): The anagram Tendre caresse témoigne, ration-elle-ment, de la jalousie de René Descartes envers Fermat.

English meaning: The anagram Tender caress testifies, [rationally] "reason-that-lies", of the jealousy of "René Descartes" towards Fermat.

In Latin the "i" was sometimes written "j" (just as the "u" was sometimes written "v"). In the space left empty at the end of his note, Fermat would have had exactly room for once to write a premonitory anagram under the words:

"i demonstrationem mirabilem sane detexi" →
"j’immortalisai anxiétés de dénombrement" (in French). In English: "I immortalized counting anxieties", but jealous people, like Descartes, would have laughed at this ‘Gascon’, this ‘braggart’...

Some anecdotes about the theorem

The most romantic of stories relates that Paul Friedrich Wolfskehl (1956-1906), a doctor, had fallen in love with a very pretty woman, but that she rejected his advances. Desperate he decided to commit suicide, fixed the day, and the hour. He put his affairs in order before the big departure, wrote letters and will. The last day arrived. As he had time left before the fateful hour, he took the opportunity to study Kummer's calculations, which explained why Lamé and Cauchy had failed in their attempt on the Fermat. Believing he had discovered a flaw in the presentation, he diligently set to work to try to fill it, but finally realized that the reasoning was sound. Dawn was already here, midnight had passed, suicide time too. But how beautiful mathematics is! He finally gives up on his disastrous project. This story is one of those that run about Paul Wolfskehl. What is certain is that suffering from multiple sclerosis he had to give up his career as a doctor and turn to mathematics. His doctorate in his pocket (probably obtained in 1880) he became interested in Fermat. After the publication of the new Diophantus by Samuel in 1670, mathematicians, captivated by the deceptive simplicity of the theorem, had indeed begun to be fascinated by it. The more time passed, the more progress they made, but there was still no proof. Two centuries passed and it was persuaded that Fermat could not have found. Paul Wolfskehl, to whom mathematics had restored the will to live, decided to pay homage to the theorem which had saved his life by offering a prize of 100,000 marks to whoever demonstrated it.
It has been reported that Euler asked a friend to search Fermat's house for clues to his proofs, including that of the theorem.
In the spring of 1994, while Wiles was still trying to close the loophole in his case and the case was looking at its worst, an Internet message ran on computer screens around the world:

Date: April 03, 1994
Subject: Fermat again!
"Today there was an extraordinary development in Fermat's Last Theorem.
Noam Elkies provided a counterexample, so Fermat's Last Theorem is ultimately wrong! [Etc.] ".

Noam Elkies, professor at Harvard, had already found at the age of 22, in 1988, a counter-example to Euler's conjecture, thus proving that it was false. The Internet message was a terrible blow to Wiles, so he would never be able to put things right! Elkies was bombarded with questions without getting any explanation. Then a mathematician took a close interest in this statement and it was realized that if the message was indeed dated April 3, it was because it had been received third-hand. The message was originally dated April 1^st, this toxic April Fool's joke had been imagined by the mathematician Henri Darmon. It did, however, have a salutary effect; the hypotheses and gossip were stopped and Wiles, Taylor, the theorem and the proof were left alone for a while.

They said

1650 (around). René DESCARTES: "Mr. Fermat is a Gascon. I, no."
1656. Blaise PASCAL, Les Provinciales (XII^th one): "All the efforts of violence cannot weaken the truth, and only serve to raise it further."
1845. Guglielmo LIBRI: "Mathematicians who had made vain efforts to prove the theorems found by Fermat wanted to cast some doubt on the reality of the demonstrations he claimed to have, and they supposed that this great geometer had arrived at certain results rather by induction and a little by chance than by a rigorous analysis of the question."
Eric Temple BELL: “In mathematics, 'obvious' is the most dangerous word."
Around 1857, there was reported a derogatory remark by Ernst KUMMER towards the great theorem, which is said to be "a joke".
1979. Ian STEWART: “Modern mathematicians have a hard time believing that Fermat could have known something that still eludes them - although, for my part, that would not surprise me."
1993. Jean BÉNABOU to Jacques ROUBAUD, after the (premature) announcement of the discovery of a proof by Andrew WILES: "It's as if we had conquered Everest with NASA rockets".
2001. Andrew WILES: "I think he was wrong when he said he had proof."
2002. G. SOUBEILLE in P. FÉRON, Pierre de Fermat, a European genius: "[…] Fermat, who was passionate about everything and kept this ambition of an encyclopedic knowledge specific to the minds of the previous century, was one of our last. humanists […]; in a broader sense, the humanist, in him, reflected his confidence in reason and in the future of science. Much more a mathematician in geometry, than a poet, he was shaped by Latin rigor and intelligence: it was on this soil that his prodigious genius of mathematics could flourish."
2009. Roland FRANQUART: "As this proof of Fermat is no longer necessary today, was it sufficient in its time?"
2020. Edwy_PLENEL on the radio France culture : “God knows if I am nevertheless warned to say that there can be great academic imbecilities, of people who are covered with diplomas as so many certainties.”
2022. Claude Mariotti: “This enigma was exactly what I needed.”

Thanks

Thanks to Roland Fanquart, of course.
A huge and sincere thanks to Wikipedia, where I could largely find out, this greatly enriched and motivated me. The numerous sources I found there, as well as on Wikisource, have been a great resource to find, sometimes very easily, the first arguments.
To Catherine Goldstein, my eternal gratitude, for the enlightenment that her work as a researcher and historian has brought me, for warm exchanges and for her encouragement. Her work, sometimes difficult, is magnificent, the study is very well documented, and above all the analyses are of great depth. The book is unfortunately out of print and difficult to find.
Thanks to Professors Emmanuel Bury and to Ludivine Goupillaud, for her study on the use of Latin at Pierre Fermat, and for his encouragements in 2009.
Thanks to Laurent Hua and Jean Rousseau for the fine remarks in their book.
Thanks to Aurélien Alvarez and Albert Violant I Holtz for their total objectivity.
Thank you all who encouraged me in this approach.
Thanks to Alexander Grothendieck, his testimony is so strong, his ideas are so deep...
Claude Mariotti

Summary bibliography

Catherine Goldstein, Un théorème de Fermat et ses lecteurs, 1995 (out of print).
Paul Tannery, Œuvres de Fermat, T. 1 (1891), 2 (1894) and 4 (1912).
Ludivine Goupillaud, ‘’Demonstrationem mirabilem detexi : mathématique et merveille dans l’œuvre de Pierre de Fermat‘’, in ‘’Tous vos gens à latin – Le latin langue vivante, langue savante, langue mondaine (XIV^e-XVII^e siècles). Ed. Droz, 2005. ISBN 2600009752.
Eric Temple Bell, The Last Problem, Ed. Simon and Schuster, 1961.
Laurent Hua and Jean Rousseau, Fermat a-t-il démontré son grand théorème ? L’hypothèse “Pascal”, Essai. L’Harmattan, 2002. The first (128 pages) is a historical study: partial formulations and their context.
Simon Singh, Fermat's Enigma, for the uninformed reader who wants to discover the history of the theorem, it is a very pleasant and interesting reading.
Albert Violant I Holtz, L’énigme de Fermat – trois siècles de défi mathématique , 2013. A collection presented par Cédric Villani.
Alexander Grothendieck, RÉCOLTES ET SEMAILLES (Reflections and testimony on a mathematician's past), a text on which every researcher could advantageously meditate.
Alexander Grothendieck, LA CLEF DES SONGES, another captivating text, in which he reveals, with uncommon conviction and strength the method that allowed him to resist prejudices, wrong beliefs and most academic opinions.
Jacques Roubaud, “Mathématique :” (narrative), Seuil, 1997.

References

↑ "In our knowledge of things in the Universe (whether mathematical or otherwise), the renovating power within us is nothing but innocence. It is the original innocence that we all shared when we were born and that rests in each of us, often the object of our contempt, and our most secret fears. It alone unites the humility and the boldness which make us penetrate to the heart of things, and which allow us to let things penetrate us and permeate us. (Harvests and sowing, p 51).
↑ De l’esprit géométrique et de l’art de persuader
↑ See also: Serendipity.
↑ Pierre de Fermat (1679). VARIA OPERA MATHEMATICA · D. PETRI DE FERMAT, SENATORIS TOLOSANI. https://documents.univ-toulouse.fr/150NDG/PPN075570637.pdf.
↑ Paul Tannery : Œuvres de Fermat - I - part 2,(p. 290-358), 1891

[1] "In our knowledge of things in the Universe (whether mathematical or otherwise), the renovating power within us is nothing but innocence. It is the original innocence that we all shared when we were born and that rests in each of us, often the object of our contempt, and our most secret fears. It alone unites the humility and the boldness which make us penetrate to the heart of things, and which allow us to let things penetrate us and permeate us. (Harvests and sowing, p 51).

[2] De l’esprit géométrique et de l’art de persuader

[3] See also: Serendipity.

[4] Pierre de Fermat (1679). VARIA OPERA MATHEMATICA · D. PETRI DE FERMAT, SENATORIS TOLOSANI. https://documents.univ-toulouse.fr/150NDG/PPN075570637.pdf.

[5] Paul Tannery : Œuvres de Fermat - I - part 2,(p. 290-358), 1891

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