641 = (2 * 320) + 1
;
then reconsider n = 641 as natural odd number ;
A1:
256 + 64 = 320
;
A2: 3 * 3 =
(3 |^ 1) * 3
.=
(3 |^ 1) * (3 |^ 1)
.=
3 |^ (1 + 1)
by NEWTON:8
;
A3:
(3 |^ 2) * (3 |^ 2) = 3 |^ (2 + 2)
by NEWTON:8;
A4:
(3 |^ 4) * (3 |^ 4) = 3 |^ (4 + 4)
by NEWTON:8;
6561 = (10 * 641) + 151
;
then
3 |^ 8,151 are_congruent_mod 641
by A4, A3, A2;
then
(3 |^ 8) * (3 |^ 8),151 * 151 are_congruent_mod 641
by INT_1:18;
then A5:
3 |^ (8 + 8),22801 are_congruent_mod 641
by NEWTON:8;
22801 = (35 * 641) + 366
;
then
22801,366 are_congruent_mod 641
;
then
3 |^ 16,366 are_congruent_mod 641
by A5, INT_1:15;
then A6:
(3 |^ 16) * (3 |^ 16),366 * 366 are_congruent_mod 641
by INT_1:18;
A7:
183,183 are_congruent_mod 641
by INT_1:11;
732,91 are_congruent_mod 641
;
then
732 * 183,91 * 183 are_congruent_mod 641
by A7, INT_1:18;
then
(3 |^ 16) * (3 |^ 16),91 * 183 are_congruent_mod 641
by A6, INT_1:15;
then A8:
3 |^ (16 + 16),91 * 183 are_congruent_mod 641
by NEWTON:8;
16653 = (26 * 641) + (- 13)
;
then
16653, - 13 are_congruent_mod 641
;
then
3 |^ 32, - 13 are_congruent_mod 641
by A8, INT_1:15;
then
(3 |^ 32) * (3 |^ 32),(- 13) * (- 13) are_congruent_mod 641
by INT_1:18;
then A9:
3 |^ (32 + 32),169 are_congruent_mod 641
by NEWTON:8;
then A10:
(3 |^ 64) * (3 |^ 64),169 * 169 are_congruent_mod 641
by INT_1:18;
28561 = (44 * 641) + 357
;
then
28561,357 are_congruent_mod 641
;
then
(3 |^ 64) * (3 |^ 64),357 are_congruent_mod 641
by A10, INT_1:15;
then
3 |^ (64 + 64),357 are_congruent_mod 641
by NEWTON:8;
then A11:
(3 |^ 128) * (3 |^ 128),357 * 357 are_congruent_mod 641
by INT_1:18;
A12:
119,119 are_congruent_mod 641
by INT_1:11;
1071,430 are_congruent_mod 641
;
then
1071 * 119,430 * 119 are_congruent_mod 641
by A12, INT_1:18;
then
(3 |^ 128) * (3 |^ 128),430 * 119 are_congruent_mod 641
by A11, INT_1:15;
then A13:
3 |^ (128 + 128),3010 * 17 are_congruent_mod 641
by NEWTON:8;
A14:
17,17 are_congruent_mod 641
by INT_1:11;
3010 = (4 * 641) + 446
;
then
3010,446 are_congruent_mod 641
;
then
3010 * 17,446 * 17 are_congruent_mod 641
by A14, INT_1:18;
then A15:
3 |^ (128 + 128),446 * 17 are_congruent_mod 641
by A13, INT_1:15;
7582 = (12 * 641) + (- 110)
;
then
7582, - 110 are_congruent_mod 641
;
then
3 |^ 256, - 110 are_congruent_mod 641
by A15, INT_1:15;
then
(3 |^ 256) * (3 |^ 64),(- 110) * 169 are_congruent_mod 641
by A9, INT_1:18;
then A16:
3 |^ 320, - 18590 are_congruent_mod 641
by A1, NEWTON:8;
A17:
- 18590 = ((- 30) * 641) + 640
;
A18:
640, - 1 are_congruent_mod 641
;
- 18590,640 are_congruent_mod 641
by A17;
then
- 18590, - 1 are_congruent_mod 641
by A18, INT_1:15;
then
ex a being natural number st a |^ ((n - 1) / 2), - 1 are_congruent_mod n
by A16, INT_1:15;
hence
641 is prime
by Th40, Th37; verum