Binomial theorem and odd power

Abstract

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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

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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

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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

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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

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Propositions

 

Proofs here for (10) p=±1[2n] and (8) p=1[2n] on math.stackexchange.com/

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  

Fermat had discovered that   and   had   prime factors (cf letters to Mersenne and Frenicle in 1640)

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: