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Binomial theorem and odd power

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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: