Abstract
This paper deals with binomial and odd power . It presents two ways of grouping terms so that is always a sum of 2 coprime numbers: a first form , and a second notable one with squares . Finally with prime, we show that prime factors are congruent to , whereas congruent to .
Introduction
We have searched how a powered number could systematically be shared into a sum of 2 coprime numbers. From binomial, we have studied different ways of grouping terms together so that . With odd and coprime of opposite parity, we have found out two possibilities. They involve the same functions that we must now introduce.
Definition
Let us define functions as
Example
Algebraic properties
Propositions
Proof
Binomial theorem gives:
Here is odd. So (1) is simply obtained by grouping together the odd power of and
(2) is a consequence of (1).
Indeed it gives
Thus by multiplying:
And finally
Which leads to the proposition by replacing
Examples for (2):
Examples in :
Proposition
Proof
(1) implies (3)
(4):
So
Coprimality
Let us consider a more detailed form of :
Proposition
Proof
First, so of opposite parity implies and odd.
The rule on gcd, , immediately implies (6) and (7).
Indeed, .
And for , so
Assertion (5) needs more attention.
Let us consider a common odd prime divisor.
The second form gives us , thus
According to the definition of
Thus , and the same
Every divisor of and divides and
Prime factors
Conjecture
Note
Fermat theorem gives and . But what a surprise to discover that it also applies to all the prime factors! And much more specifically on the
Let us remind the Fermat's theorem on sums of two squares:
And the Euler's theorem: , which is here
Let us note that these also appear in Fermat-Wiles theorem with (3)
Examples for
Examples for . The number of factors is even
Examples with both squared variables: