The Wieferich numbers up to 1015 :
1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, 1232361, 2053935, 2685501, 3697083, 3837523, 6161805, 11512569, 18485415, 19187615, 26862661, 34537707, 49887799, 57562845, 80587983, 103613121, 134313305, 149663397, 172688535, 241763949, 249438995, 310839363, 349214593, 402939915, 448990191, 518065605, 648541387, 725291847, 748316985, 1047643779, 1208819745, 1346970573, 1554196815, 1746072965, 1945624161, 2175875541, 2244950955, 3142931337, 3242706935, 3626459235, 4040911719, 4539789709, 5238218895, 5836872483, 6527626623, 6734852865, 9428794011, 9728120805, 10879377705, 12122735157, 13619369127, 15714656685, 17510617449, 20204558595, 22698948545, 28286382033, 29184362415, 32638133115, 40858107381, 47143970055, 52531852347, 60613675785, 68096845635, 84859146099, 87553087245, 122574322143, 141431910165, 157595557041, 204290536905, 254577438297, 262659261735, 367722966429, 424295730495, 612871610715, 787977785205, 1103168899287, 1272887191485, 1838614832145, 3309506697861, 5515844496435, 16547533489305.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 104 values, from 1093 to 16547533489305).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 0 | 104 | 2 | ||||||||
| 3 | 84 | 12 | 8 | 3 | |||||||
| 4 | 0 | 52 | 0 | 52 | 4 | ||||||
| 5 | 46 | 14 | 14 | 15 | 15 | 5 | |||||
| 6 | 0 | 12 | 0 | 84 | 0 | 8 | 6 | ||||
| 7 | 47 | 10 | 7 | 10 | 11 | 10 | 9 | 7 | |||
| 8 | 0 | 26 | 0 | 26 | 0 | 26 | 0 | 26 | 8 | ||
| 9 | 64 | 4 | 3 | 12 | 5 | 3 | 8 | 3 | 2 | 9 | |
| 10 | 0 | 14 | 0 | 15 | 0 | 46 | 0 | 14 | 0 | 15 | 10 |
| 11 | 0 | 12 | 10 | 10 | 10 | 12 | 12 | 8 | 9 | 10 | 11 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.
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