Fermat's small theorem de Fermat (condensed):

In mathematics, the small theorem of Fermat 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.

Remember : prime of Mersenne :

if then is a prime one?

no : because it's don't work for 2¹¹

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)

Comments

Matrice showing this guess (a=2)

Fermat small theorem don's separate prime and not prime as well as wish.:

Samples :

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

nb: 3 as 2 is prime...

n	3⁶	3⁶-3	726/6	Rem
6	729	726	121	0

pour a=5 , n=10

n	5¹⁰	5¹⁰ -5	5¹⁰ -5/5	Rem
10	9765625	9765620	976562	0

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

For a= 4 it's worse...

n	4ⁿ	4ⁿ-4	4ⁿ-4/4	r
2	16	12	6	0
3	64	60	20	0
4	256	252	63	0
5	1024	1020	204	0
6	4096	4092	682	0
7	16384	16380	2340	0

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

n	2ⁿ	2ⁿ– 2	2ⁿ– 2 / n	mod
1	2	0	0	0
2	4	2	1	0
3	8	6	2	0
4	16	14	3 1/2	1/2
5	32	30	6	0
6	64	62	10 1/3	1/3
7	128	126	18	0
8	256	254	31 3/4	3/4
9	512	510	56 2/3	2/3
10	1024	1022	102 1/5	1/5
11	2048	2046	186	0
12	4096	4094	341 1/6	1/6
13	8192	8190	630	0
14	16384	16382	1170 1/7	1/7
15	32768	32766	2184 2/5	2/5
16	65536	65534	4095 7/8	7/8
17	131072	131070	7710	0
18	262144	262142	14563 4/9	4/9
19	524288	524286	27594	0
20	1048576	1048574	52428 5/7	5/7
21	2097152	2097150	99864 2/7	2/7
22	4194304	4194302	190650 1/9	1/9
23	8388608	8388606	364722	0
24	16777216	16777214	699050 4/7	4/7
25	33554432	33554430	1342177 1/5	1/5
26	67108864	67108862	2581110 1/9	1/9
27	134217728	134217726	4971026 8/9	8/9
28	268435456	268435454	9586980 1/2	1/2
29	536870912	536870910	18512790	0
30	1073741824	1073741822	35791394 1/9	1/9
31	2147483648	2147483646	69273666	0
32	4294967296	4294967294	134217727 8/9	8/9
33	8589934592	8589934590	260301048 1/6	1/6
34	17179869184	17179869182	505290270 1/9	1/9
35	34359738368	34359738366	981706810 4/9	4/9
36	68719476736	68719476734	1908874353 5/7	5/7
37	137438953472	137438953470	3714566310	0