# Fermat's enigma decrypted

This resource includes primary and/or secondary research. Learn more about original research at Wikiversity. |

This study is much more complete on fr.wikiversity

“*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)

‘‘*I had to work to discover them. Do work, you too, you will thus realize that it is in this work that the majority of the pleasure consists.*’’ (Pierre de Fermat, around 1646).

Fermat's Last Theorem is the name given to Fermat's ultimate challenge.

**x ^{n} + y^{n} = z^{n}**,

**impossible when n is an integer greater than 2.**

## Foreword[edit | edit source]

*“Do you want to know what is most honorable? Do Love to be ignored.”* (Final words of Fermat's epitaph).

This essay is constantly evolving. The study of Fermat's correspondence requires a great deal of time, and as soon as one becomes attached to it one does not cease making new discoveries. Understanding the psychology of such a character in an attempt to discover all that he wanted to mean by his tricks is an endless job. It is only through discoveries that we make (we go from surprise to surprise) and at the cost of long meditations that we can progress in this work. This work is difficult, because the collective imagination is there, which constantly reminds us the final judgment that very great scholars made against Fermat.

## Fermat on the psychoanalyst’couch (smoke and mirrors?)[edit | edit source]

The analysis of his second *OBSERVATIO* (about *QUÆSTIO VIII* of *Arithmetica)*, the study of his works, his correspondence, his life and especially the psychology of the person, is a subject of constant meditation. In the form of an enigma, this observation is a treasure of ingenuity, the acme of the entire book that Fermat wanted to dedicate to the science of numbers. This book, which was supposed to extend far away, ‘’in an astonishing way’’', the bounds of the science of numbers, is it sorely lacking? The great work of Fermat consists mainly of:

– 47 comments marked "OBSERVATIO D.P. F.",

– 1 comment marked “OBSERVATIO DOMINI PETRI DE FERMAT” (the enigmatic one).

These 48 comments that would take 10 or 12 pages were added by his son Samuel to the Arithmetica of 1621, to compose the *Arithmetica* of 1670. This is a new book that contributed a lot to the knowledge, a book the prologue of wich is longer than Fermat's text. Over time this observation of the seventeenth century was roughly translated into different versions to which mathematicians have always relied, since only the principle of the theorem, which was perfectly stated in it, seemed important to them.

Around 1640, a little enlarged, here is a photograph of Fermat's comment, transcribed later on the Arithmetica of 1670. It's the edition at Lyon Library , which drew the attention of Mr. Franquart in 2009: at the end of the third line, the word *detexi*: when we look at it with a magnifying-glass (third photo), the overprinted letter “* t”* looks like a stain that would have made with meticulousness: in the context and without enlarging the picture, it does't draw the attention of a reader who doesn't look for a clue, for anything unusual, incongruous.

The point following the word *detexi* is overloaded on 3 different versions of the same edition. Here is the literal translation of this ‘Observatio’, that Fermat intended for the reader who will take him for a boastful, an "amateur" (a more elaborate version, to the attention of the researcher, after decoding by Mr. Roland Franquart, is available on his website).

“*But whether it would be a cube into two cubes or a fourth power into two fourth powers and generally up to infinity, no power greater than the square can be divided into two like powers. I really have laid bare a surprising demonstration [explanation] of this, the margin could not contain it.*”

In modern terms: “No three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2.”

### Demonstrationem mirabilem sane detexi[edit | edit source]

Let's have a close look at this phrase, worded in a very special way.

1. First of all, note that *detexi* means ‘’I have laid bare’’, ‘’I removed what covered’’, which may easily be confused with ‘’I discovered’’ or with “I found”.

2. The order of the words.

– *Cuius rei mirabilem demonstrationem sane detexi*, would have been a correct phrase:

“What I really brought out the surprising explanation”.

– *Cuius rei sane mirabilem demonstrationem detexi *, would also have been correct:

“What I brought out the really surprising explanation.”

Fermat didn’t not use any of these formulations, why? If the Latin language may seem complex, it is a rigorous, concise language, which has very precise rules. Thus, this rigor had an advantage for the Latins, to derogate to them allowed to say things while hiding them. They often used this resource.

(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 reasonings. [...]’’. 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).

In Fermat's formulation, if “sane” and “detexi” are in the right order (adverb before verb), the 2 preceding words are not in order, since the adjective (*mirabilem*) has to be placed before the name (there are some rare exceptions). Like in his letter to Carcavi, Fermat formulates in a curious way, where the adverb *really* is addressed not only to “I laid bare” but also to “marvellous”. In an *Observation* already surprising, since he says the margin is too small, and since we consider the curiosities already mentioned, it is a new curiosity. If one wants to translate literally, to the address of the reader who doesn’t linger, the correct translation is:

- What I have truly laid bare the really surprising explanation.

Then taking into account the decoding of Roland Franquart:

- What I have truly weaved, fully, the really surprising explanation.

We think Fermat was sure that the sentence, badly translated, would spoil the followers of the followers, who would see him as a joker and a boaster.

### About Fermat numbers[edit | edit source]

Fermat submitted this conjecture to 7 of his correspondents, asking them to kindly help him to prove it. Using prime numbers *74k+1*, Fermat had already found that 2^{37} – 1 (*ie* 137,438,953,471) is divisible by 223. Using a similar imethod, divisors *64k + 1*, he would have immediately shown, with the help of 4 short divisions, that *F*_{5} is divisible by (64×10) + 1 (*ie* 641), and therefore is not prime. Fermat has always said he had no proof of this proposal – which does not appear on *Arithmetica*, where all his proposals have been rigorously proved.

Catherine Goldstein, in her book *Un théorème de Fermat et ses lecteurs*, studies in particular Fermat's Observation XLV (its formulation, readings, etc.), which is the only complete proof of a theorem that he revealed (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 his 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^{[1]} [published by her 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 (Fermat numbers’conjecture). It is noted in these Varia Opera (Paul Tannery had noted it) that the first letter from Frenicle de Bessy is misses as well. It is not the only one, because apart from one, all Fermat's letters dealing with false conjecture are missing:

1) to Frenicle de Bessy, ‘August’ (?) 1640, where these words appear, the context of which Fermat them 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. Eventually it's this ambiguous formulation that will become after his death the most famous of his remarks on these

*Fermat numbers*. The formulation of this conjecture is unusual, it begins like this:

*“I then*, then towards the end:

**considered**certain questions”*“All squared powers of 2 augmented by 1 are prime numbers.”*

In 1977, Harold Edwards wrote:

*“He [Fermat] went so far as to say, later in his life, that he could prove that these numbers were all prime”*– which is far from obvious to Eric Temple Bell

^{[2]}, who makes the remark. But Edwards continues:

*“I do not see any other interpretation of this letter to Carcavi.”*Why do many mathematicians may read : “Then I proved that ...”? To have a very good argument and say that Fermat was not reliable? Anyway, Bell did not share this opinion. The fact that Fermat gives us the choice between two options should be noted. Eventually it's this formulation for Huygens that will become after his death the most famous of his remarks on these

*Fermat numbers*. Note also that he no longer uses the word “proposition(s)” as he does elsewhere: “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. Here are yhe next ones: [...] “All squared powers of

*2*augmented by

*1*are prime numbers. This last question is of a very subtle and very ingenious search and, although it is conceived affirmatively, it is negative, since to say that a number is prime, it's to say that it can not be divided by any number.” Huygens was a young 30 years old scientist and mathematician, the only one who could have followed him, but he did not follow answered. The formulation of this last test balloon was even so very exciting, Fermat increased the intelligence of this research by adding to the word “subtle" its synonym, “ingenious”.

This letter to Carcavi is the the only known text written by Fermat where he uses the expression “question (s) negative (s)”. Is there a double meaning here, implying that the answer is negative? moreover fhe expression question(s) negative(s) is not very correct, especially from a philologist, especially in a balance sheet. A question is always a question, it cannot be negative. These letters were written over a period of 19 years to ask for help (...). Have they been written for the sole purpose of testing its correspondents and/or to stimulate them? The other hypothesis is that his research path was quite different and that he did not think to use prime numbers of the *64k+1* form as divisors, which woluld be rather surprising. Anyway, this letter will be an argument for skeptical commentators about Fermat's competence.

CM) These *VARIA OPERA MATHEMATICA* (various mathematical works) are a selection of articles and letters. When Samuel published them after his father's death (as he did for the *bservations*), he inserted a single letter evoking the false conjecture, the one addressed to Monsieur de ****. We are almost 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, of which, 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 [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. His commentators never wondered (to our knowledge) why Samuel Fermat published this single wording. On the contrary, 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 Huygens was his last attempt to find someone to sahre with.

Let's quote once more Émile Brassinne (1853, "Bnf", "Gallica") in his *Mathematical works'guidebook of P. Fermat and Arithmetic by Diophantus*

### ‘Lack of space’[edit | edit source]

(CM, Jean Rousseau, Laurent Hua). According to Samuel Fermat, his father wrote these observations in the margins of a copy of *Arithmetica*. This copy disappeared, nobody has been astonished. Is it really on this Arithmetica that all these observations were *totally* written? Some of them (n°6, n°7), are very long and could hardly have kept in a margin. If Fermat has given, in a libretto or on free paper, precise instructions to his son in the manner of writing this note so important to him, in three different editions, these instructions justify the disappearance of a work of a considerable historical value, a work that Samuel would have been obliged to destroy. Obviously, Fermat had asked his son to make his 48 observations known after his death. They were written in an irreproachable style, perfectly legible. A margin is not the best way to transmit them, especially if they are accompanied by instructions.

## The style of the 48 *observations*[edit | edit source]

- (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 had 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?

## Pascal's triangle[edit | edit source]

*“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 comfor 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 strived so much 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.

What attitude could have a contemporary scientist vis-a-vis 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 would be undemonstrable, 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 seemded 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 hab 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 urnan 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 throwing a challenge to the world.

## The urban legend[edit | edit source]

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 argumentst 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 (I presume it has not many ‘sisters’), 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 letter-review 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, 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!nThis may 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.”[2].

When studying Fermat, there are two ways of proceeding:

1) With an *a priori* favorable, always remembering that he is a pedagogue, then trust him and detect every argument obviously spécious that his detractors have written; on the contrary, seek the many clues he leaves us, and all good arguments (I counted fourteen important ones). It may be noted that all the authors who have devoted a complete work to the last theorem, with the only interest in informing the reader, always remained objective and neutral.

2) With an *a priori* openly unfavorable: thus, we underestimate him, without trusting his dearest and most commendable desire to facilitate our work. We then imagine multiple arguments to discredit him.

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 Euclide'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. think (I have really laid bare the quite surprising explanation [but, be happy!] there is no room here).

.../...

(voir brouillon MS)

## Three different versions of Arithmetica: first codings/indices[edit | edit source]

– During more than three centuries, scientists have never read a faithful translation of Fermat's second *OBSERVATIO*: “I really have laid bare 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 exposed*, or *took away what covered*, *unveiled*.

– In his *Works'guidebook of P. Fermat and Arithmetic by Diophantus* (Toulouse, 1853), Émile Brassinne delivers the best translation that we can find: he put the word *really* at the right place, not writing, like many did, *really marvellous*. However, this translation is not perfect, as he translates *I really found* (‘found’ would be *inveni* in Latin), which helps the reader to drive away from a in-depth study of the OBSERVATIO. In many *translations* that can be found, the adverb *really *is out of place, it does not apply to *demonstrationem* , but to the author: "I really have laid bare".

– 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 touhgt me, in 2009, his research about the "Observation" present on the *Arithmetica* at Lyon Library, my passion for this enigma (which the treatment seemed to me shocking) was further increased. In June 2017, I have been searching for a long time another quirk, still on the word *detexi,* that could have appeared on another version of *Arithmetica*, where Roland Franquart had already found a quirk (the** t **overloaded, version B). Frankly, I did not believe it at all (it would have been ‘‘too much’’, even from Fermat). If I was so persevering, it's that deep inside me I wanted to find a clincher. Finally I found another quirk on

*Arithmetica*at Rome University (the

*ṡ*). I hardly believed my eyes, this discovery, quite strange, left me so thunderstruck and dubious (the clincher, I received it myself!). It took 18 months to me to find what it meant (once on the site Web, mount the pointer to the top, a black horizontal band appears, enter the page number 141 (which is page 61 of Book II), then enlarge the image (sign + on the bottom):

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, active voice, indicative mood → “you weave completely”^{[3]}, (or “you represent completely”, “you arrange in braids”, “you finish a fabric”), which joins and confirms the alphanumeric decryption performed by R. Franquart in 2009.

– Zurich Library, le word is correctly written:

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]” whose prologue (Diophantus) is longer than the book itself (Fermat), 340 pages against a fortnight. The proof really “laid bare” 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, tectum*) his explanation in a masterful feat.

## Codings common to each version[edit | edit source]

Fermat's tips are remarkable. Thanks to R.F. who discovered the most important clues and informed me in 2009, as he had learned this enigma fascinated me. I repeat here the most symbolic, sometimes modifying them somewhat (I hope not to betray too much his thought), and I 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**, for Rome edition in particular (*detex ṡ *). The two oddities on the letters

*t*and

*u*, direct us to the choice of these two letters to attempt a weaving, and this begins to be confirmed since in its note we find 21

*u*(

*u*is the 21

^{e}letter of alphabet), and only 19

*t*(

*t*is the 20

^{e}), because of

*not caperet*= would not contain → this famous

*, whose importance is still underlined by the overloaded dot; these coincidences between the number of letters*

**t***u, t,*and their respective rank in alphabet strongly incite us to choose them as elementary bricks of an encryption; then we see that in his text, Fermat using the Latin letters

*u*and

*v*, equivalent in Latin, succeeded in introducing in the first part of the text two 'consecutive' couples of contiguous letters

*ut*, and then 3 consecutive couples

*tu*to perform his weaving (the Latin letters

*u*and

*v*being equivalent in Latin, he spells two others words, in his observation, in a rather personal way) (R.F. site).

– (R.F.) Fermat succeeds in introducing in the first part of his text two consecutive couples "ut", and then three consecutive couples "tu" (which means "you" in French) to carry out his weaving.

– (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 square into two squares of square) refers to the case

*n=4*, whose proof appears in the only theorem Fermat has fully explained. It's surely a real coincidence here, that is timely.

Across the centuries, many scholars have doubted that Fermat really had 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'sorks.

Considering the possible existence of a coding in a Fermat's note, the one described by Roland Franquart seems to be such an impossible feat, even coming from Pierre de Fermat, that we could think: “No, this must me be coincidences, even a genius could not ferform such an impossible feat”. Perhaps this feat would be simply masterful?

## Amazing anagrams[edit | edit source]

In French:

- Pierre de Fermat préféra méditer. Petri de Fermat permettra défi dernier théorème, étreindre Homère.

English translation:

- Pierre de Fermat prefered to meditate. Petri de Fermat will allow challenge last theorem, embrace Homer.

## Thanks[edit | edit source]

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.

All my gratitude to Catherine Goldstein for the lighting that his work brought me, for cordial exchanges and for her encouragements. Her book, sometimes arduous, is magnificent, well documented, and above all, the analyzes are of great mathematical depth. The book is unfortunately exhausted.

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 Alexandre Grothendieck, his testimony is so strong, his ideas so deep...

And thanks to Hor des Fields de Mauny, my faithful Golden Retriever, who accepts, sometimes with difficulty, that I'm not always playing with him.

## Restricted bibliography[edit | edit source]

- Paul Tannery,
*Œuvres de Fermat, T.*1 (1891), 2 (1894) and 4 (1912).

- Catherine Goldstein,
*Un théorème de Fermat et ses lecteurs*, 1995 (out of print).

- 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,
Ed. Simon and Schuster, 1961.**The Last Problem**

- 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 présented par Cédric Villani.

- Alexandre Gothendieck, RÉCOLTES ET SEMAILLES (Reflections and testimony on a mathematician's past), a text on which every researcher could advantageously meditate.

- Alexandre 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, false beliefs and most academic opinions.

- Jacques Roubaud :
*“Mathématique :”*(narrative), Seuil, 1997.

## Notes[edit | edit source]

- ↑ Pierre de Fermat (1679).
*VARIA OPERA MATHEMATICA · D. PETRI DE FERMAT, SENATORIS TOLOSANI*. https://documents.univ-toulouse.fr/150NDG/PPN075570637.pdf. - ↑ ‘’E. T. Bell predicted that civilisation would come to an end as a result of nuclear war before Fermat's Last Theorem would ever be resolved.’’ (Simon Singh)
- ↑ Online Latin-English Dictionary:
*detexis*is a conjugate form → ‘’You weave completely’’