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 ap -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), 1p
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  + [p!-(1+(p-1)!] + [(p-1)!] , 0 (mod p) , 1+(p-1)! is divisible par p .
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) = [1np]+[anp-1+bnp-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 ( np -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 2n-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 !!!:
Matrice showing this guess (a=2)
Fermat small theorem don's separate prime and not prime as well as wish.:
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
by add one on two sides
Is 221 dividable by 7?
221 n'est pas divisible par 7
(31*7 reste 4)
Is 91 dividable by 7?
Patrick Stoltz le 18/10/2009