Fermat's little  theorem (condensed):

In mathematics, the Fermat's little theorem is a result of the modular arithmetic, which can also demonstrate itself with the tools of the elementary arithmetic.

It expresses itself as follows. If 'a' is a not divisible integer by p such as p is a prime number, then a ^p-1 -1 is a multiple of p. The corollary of this theorem is that, for every 'a' interger and 'p' prime number, then a^p -1  is a multiple of p.

He owes his name to Pierre de Fermat (on 1601 - 1665) who expresses it the first time on October 18th, 1640.

the Fermat's poly polygonals "key" formula

Example for =

Fermat has proved all his theories (little ,and last theorem) for all power of n , so it is absolutely excluded that Fermat have built them by EACH indices on n. example for 7! indices are 1,21,175,735,1634,1764,720 (see before) . But we can , with a small trick, found the sum off all of them.

How to find the sum of all indices ?

If ,we shift n by 1 (n=1) then we get 

If we develop n(n+1)(n+2)...(n+p) , the sum of all indices is equal of the factorial of denominator ,  example for 7! , 1+21+175+735+1634+1764+720 = 5040

Development of numerator:

if p numbers of factors of n(n+1)(n+2).. then

The FIRST exponent = numbers of factors (as Pascal's board),  1^p

The FIRST indice is equal to 1

The LAST  indice = factorial -1 of numbers of factors.

Exemple : nombre de facteurs = 7

<=>

The FIRST exponent = numbers of factors : 1n⁷

The FIRST indice is equal to 1 , 1+21+175+735+1624+1764+720=5040

The LAST  indice = factorial -1 of numbers of factors.1+21+175+735+1624+1764+720=5040 , 1 x 2 x 3 x 4 x 5 x 6 =720

Shifting n by 1

For example for 7 factors

if n = 1 then ,

1+21+175+735+1624+1764+720=5040 = 7!

Demonstration of Fermat's small theorem

Primaries numbers and factotials

If a number is primary , understand divisible only by himself or 1 , imply that only that the last factor of his factorial is divisible by this number , by example , 7 is primary so 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 only the last factor , 7 , is divisible by  7 , and NOT the others 2,3,4,5,6 (we get a NOT INTEGER one , we get a decimal number or real )

We deduct also that if a number is primary his factorial LESS ONE is not divisible by this number 2 x 3 x 4 x 5 x 6 is not divisible by 7.

Concatenation  in 3 terms , first / intermediate / last

P! is a sum of p terms which the first equal to 1 and the last one  = (p-1)! , We add all intermediate terms in a second term =

p!-1-(p-1)! = p!-(1+(p-1)!)  so 120 = 1 + [120-(1+24)] + 24 = 1 + 95 + 24

The second term is divisible by p (as show bellow) , so we can discard it..

If p is primary , then (p-1)! is not divisible by p , 1 is not divisible by p , 1+ [p!-(1+(p-1)!)] + (p-1)! is divisible by p

<=>

If [1] + [p!-(1+(p-1)!] + [(p-1)!] , 0 (mod p) , 1+(p-1)! is divisible par p .

More details

For example p=7 then 1+1*2*3*4*5*6 = 721 divisible by 7

Replace now 1 by n = get the Fermat's small theorem

If n(n+1)(n+2)...(n+p-1) = [1n^p]+[an^p-1+bn^p-2]+[(p-1)!n] then

In the last term of  1n⁵+10n⁴+35n³+50n²+24n , 24n + n = 25n is divisible by 5 , so the first - n is divisible by 5 , it means that if we adding n to the last term for make it divisible we have to subtract n to the fisrt  ( n^p -n)

 Prime  number's guess in corollary of Fermat's small theorem

n is prime  if  (or )  

Shoud redable as following : if the reminder of  2ⁿ-2 / n is 0 <=> if the reminder of 2^n-1-1 /n est 0

Ensuing of Fermat's small theorem,if a=2; if remainder (= modulo 1) of a ⁿ-a/a  = 0 then n is prime number (if not n is not prime)

It's only a guess because it's necessary to be able to demonstrate it (*),2 ⁿ Exceeds the capacities of calculations and calculators...

(I'm writing modulo 1 because it's the computer function , should be )

It's very important to see that in this guess only 2ⁿ put numbers into 2 well separed groups , primes and non primes

(*) Professor Henry Cohen of the university of Bordeaux notes " We can conclude nothing if 2n-1 1 is divisible by n, but it is rather likely in this case that n is prime (Number's history Tallandier Edition 2007)

This conjecture is not true for (n< 1000) 341,561,645 !!!:

Comments

 Matrice showing this guess (a=2)

Samples :

a=3 n=6 , 6 is not prime , the 3⁶-3/3 reminder should be <> 0

nb: 3 as  2 is prime...

pour a=5 , n=10

It's seems to be the same for all n/a = 2 ...

For a= 4 it's worse...

Small appended notes:

If we want to know if a number is divisible by other one of head you should:

If it's even divide it by 2

If it's odd substract it : "lui ôter une mesure"as would have written it Fermat....)

Ensue of binary algorithm

 .

If mod = 0 then n is prime ... if not n is not

Important notice

by add one on two sides

Is 221 dividable by 7?

221-7=214

214/2=107

107-7=100

50/2=27

27-7=20

20/2=10

10/2=5

221 n'est pas divisible par 7

(31*7 reste 4)

Is 91 dividable by 7?

91-7=84

84/2=42

42/2=21

21-7=14

14-7=7

.

Patrick Stoltz le 18/10/2009

pstoltz@shemath.com