During this study, I try to visualize prime numbers, their distributions and the means of reduire major powers of 2...To do it, I use the topology of formed squares (according to the same as topologic squares principle) to refine / counter this guess.

2ⁿeven and odd

Square topology formed by 2ⁿ are different according if n even or odd (Always and still)

To obtain an even root from even square (in binary for example) you can divide power by 2

2⁸= a square of 2⁴ * 2⁴, 256 is a square of 16*16

On the other hand for a 2 ^odd you should divide this number by 2 (or apply 2 ^n-1), then form a rectangle composed by 2 even squares witch the smallest ridge is the root of the even square, and the big one 2 * the root of even square

Example: 2⁵ = a rectangle of 2 ^(5-1)/2 * 2(2 ^(5-1)/2) or (2 ⁽⁵⁺^1)/2)

2⁵ = 2² * 2³, 32 is a rectangle of 4 * 8

(V signify OR), will be explained following

Graphing 2ⁿfly

This animation demonstrate also the Fermat's two square guess

Guess : is 2 ^n-1 -1 / n mod 1 =0 is prime / Graphing Fermat's two squares guess

By definition a prime numbers is odd , except 2, there are in very precise group.

As showed by Riemann, odd numbers are in 1/2 + it axis, (even numbers in négative axis)

If n is always odd, n-1 is thus a power always even.

If apply (for n odd), all primes numbers are drawing by a 2^{n even}topologie ,square from witch we remove 1.

Then we divide by n for know if mod = 0 and if it's prime or not

This guess demonstrate also the Fermat's two square guess: all primes divided by 4 is modulo 1

To the left : This 2⁴ square have a side of 2²


n	^n-1	^(n-1)/2
1	0
3	2	1	2	2
5	4	2	4	4
7	6	3	8	1
9	8	4	16	7
11	10	5	32	10
13	12	6	64	12
15	14	7	128	8
17	16	8	256	1
19	18	9	512	18
21	20	10	1024	16
23	22	11	2048	1
25	24	12	4096	21
27	26	13	8192	11
29	28	14	16384	28
31	30	15	32768	1
33	32	16	65536	31
35	34	17	131072	32
37	36	18	262144	36
39	38	19	524288	11
41	40	20	1048576	1
43	42	21	2097152	42
45	44	22	4194304	34
47	46	23	8388608	1
49	48	24	16777216	8
51	50	25	33554432	2
53	52	26	67108864	52
55	54	27	134217728	18
57	56	28	268435456	55
59	58	29	536870912	58
61	60	30	1073741824	60
63	62	31	2147483648	2
65	64	32	4294967296	61
67	66	33	8589934592	66
69	68	34	17179869184	25
71	70	35	34359738368	1
73	72	36	68719476736	1
75	74	37	137438953472	47
77	76	38	274877906944	25
79	78	39	549755813888	1
81	80	40	1099511627776	70
83	82	41	2199023255552	82
85	84	42	4398046511104	4
87	86	43	8796093022208	56
89	88	44	17592186044416	1
91	90	45	35184372088832	57
93	92	46	70368744177664	64
95	94	47	140737488355328	53
97	96	48	281474976710656	1
99	98	49	562949953421312	83

Modulo (rest off the strip,root divided by n) seems to be 1 or n-1

Because the square's topology always belongs = to one 2n peer for a square, I cut this square in bands(strips) and then divide it by n to find the modulo

This avoids among others the division by n of the complete square which represents hudge number

Study this asided board:

n = odd number

n-1=power of "Fermat's square" (2 ^n-1)

(n-1)/2 = power of rooted square (=side) of "Fermat's square"

= root of a square witch we apply the Fermat's small theorem. (n odd -1 is even)