Хотелось бы представить Вашему вниманию один из вариантов алгоритма факторизации составного числа.
Как уже отмечалось[1], есть закономерности распределения значений квадратичных вычетов, как для простых, так и для составных чисел.
Следует привести известную зависимость[2]. Если число A, целое, положительное, равно произведению простых чисел a и b, то всегда найдутся такие два числа c и d, что c2 – d2 = A или c2 – d2 = nA , где n целое число от 1 до ( A – 1 ). При этом,
c2 – d2 = (c + d)(c – d), т.е. (c + d) и (c – d ) кратны делителям a и b.
Для упрощения рассмотрим матрицу остатков составного числа A = 35, так же как в [1], представленную на рис.1.
34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 | 34 | 1 |
33 | 4 | 27 | 16 | 3 | 29 | 12 | 11 | 13 | 9 | 17 | 1 | 33 | 4 | 27 | 16 | 3 | 29 | 12 | 11 | 13 | 9 | 17 | 1 | 33 | 4 | 27 | 16 | 3 | 29 | 12 | 11 | 13 | 9 |
32 | 9 | 8 | 11 | 2 | 29 | 18 | 16 | 22 | 4 | 23 | 1 | 32 | 9 | 8 | 11 | 2 | 29 | 18 | 16 | 22 | 4 | 23 | 1 | 32 | 9 | 8 | 11 | 2 | 29 | 18 | 16 | 22 | 4 |
31 | 16 | 6 | 11 | 26 | 1 | 31 | 16 | 6 | 11 | 26 | 1 | 31 | 16 | 6 | 11 | 26 | 1 | 31 | 16 | 6 | 11 | 26 | 1 | 31 | 16 | 6 | 11 | 26 | 1 | 31 | 16 | 6 | 11 |
30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 | 25 | 15 | 30 |
29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 | 29 | 1 |
28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 | 7 | 21 | 28 | 14 |
27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 | 13 | 1 | 27 | 29 |
26 | 11 | 6 | 16 | 31 | 1 | 26 | 11 | 6 | 16 | 31 | 1 | 26 | 11 | 6 | 16 | 31 | 1 | 26 | 11 | 6 | 16 | 31 | 1 | 26 | 11 | 6 | 16 | 31 | 1 | 26 | 11 | 6 | 16 |
25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 | 30 | 15 | 25 |
24 | 16 | 34 | 11 | 19 | 1 | 24 | 16 | 34 | 11 | 19 | 1 | 24 | 16 | 34 | 11 | 19 | 1 | 24 | 16 | 34 | 11 | 19 | 1 | 24 | 16 | 34 | 11 | 19 | 1 | 24 | 16 | 34 | 11 |
23 | 4 | 22 | 16 | 18 | 29 | 2 | 11 | 8 | 9 | 32 | 1 | 23 | 4 | 22 | 16 | 18 | 29 | 2 | 11 | 8 | 9 | 32 | 1 | 23 | 4 | 22 | 16 | 18 | 29 | 2 | 11 | 8 | 9 |
22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 | 8 | 1 | 22 | 29 |
21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | 21 |
20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 | 20 | 15 |
19 | 11 | 34 | 16 | 24 | 1 | 19 | 11 | 34 | 16 | 24 | 1 | 19 | 11 | 34 | 16 | 24 | 1 | 19 | 11 | 34 | 16 | 24 | 1 | 19 | 11 | 34 | 16 | 24 | 1 | 19 | 11 | 34 | 16 |
18 | 9 | 22 | 11 | 23 | 29 | 32 | 16 | 8 | 4 | 2 | 1 | 18 | 9 | 22 | 11 | 23 | 29 | 32 | 16 | 8 | 4 | 2 | 1 | 18 | 9 | 22 | 11 | 23 | 29 | 32 | 16 | 8 | 4 |
17 | 9 | 13 | 11 | 12 | 29 | 3 | 16 | 27 | 4 | 33 | 1 | 17 | 9 | 13 | 11 | 12 | 29 | 3 | 16 | 27 | 4 | 33 | 1 | 17 | 9 | 13 | 11 | 12 | 29 | 3 | 16 | 27 | 4 |
16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 | 11 | 1 | 16 |
15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |
14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 | 14 | 21 |
13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 | 27 | 1 | 13 | 29 |
12 | 4 | 13 | 16 | 17 | 29 | 33 | 11 | 27 | 9 | 3 | 1 | 12 | 4 | 13 | 16 | 17 | 29 | 33 | 11 | 27 | 9 | 3 | 1 | 12 | 4 | 13 | 16 | 17 | 29 | 33 | 11 | 27 | 9 |
11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 | 16 | 1 | 11 |
10 | 30 | 20 | 25 | 5 | 15 | 10 | 30 | 20 | 25 | 5 | 15 | 10 | 30 | 20 | 25 | 5 | 15 | 10 | 30 | 20 | 25 | 5 | 15 | 10 | 30 | 20 | 25 | 5 | 15 | 10 | 30 | 20 | 25 |
9 | 11 | 29 | 16 | 4 | 1 | 9 | 11 | 29 | 16 | 4 | 1 | 9 | 11 | 29 | 16 | 4 | 1 | 9 | 11 | 29 | 16 | 4 | 1 | 9 | 11 | 29 | 16 | 4 | 1 | 9 | 11 | 29 | 16 |
8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 | 22 | 1 | 8 | 29 |
7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 | 28 | 21 | 7 | 14 |
6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 | 6 | 1 |
5 | 25 | 20 | 30 | 10 | 15 | 5 | 25 | 20 | 30 | 10 | 15 | 5 | 25 | 20 | 30 | 10 | 15 | 5 | 25 | 20 | 30 | 10 | 15 | 5 | 25 | 20 | 30 | 10 | 15 | 5 | 25 | 20 | 30 |
4 | 16 | 29 | 11 | 9 | 1 | 4 | 16 | 29 | 11 | 9 | 1 | 4 | 16 | 29 | 11 | 9 | 1 | 4 | 16 | 29 | 11 | 9 | 1 | 4 | 16 | 29 | 11 | 9 | 1 | 4 | 16 | 29 | 11 |
3 | 9 | 27 | 11 | 33 | 29 | 17 | 16 | 13 | 4 | 12 | 1 | 3 | 9 | 27 | 11 | 33 | 29 | 17 | 16 | 13 | 4 | 12 | 1 | 3 | 9 | 27 | 11 | 33 | 29 | 17 | 16 | 13 | 4 |
2 | 4 | 8 | 16 | 32 | 29 | 23 | 11 | 22 | 9 | 18 | 1 | 2 | 4 | 8 | 16 | 32 | 29 | 23 | 11 | 22 | 9 | 18 | 1 | 2 | 4 | 8 | 16 | 32 | 29 | 23 | 11 | 22 | 9 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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