A Wieferich prime is an odd prime p such that p2 divides 2p−1−1, a generalized Wieferich prime base n is an odd prime p such that p2 divides np−1−1 (by Fermat’s little theorem, all primes p not dividing n divide np−1−1).
The only two known Wieferich primes are 771 and 2047 (both of them are one greater than numbers with periodic binary (base 2) expansions (770 = 0100010001002 = 44414; 2046 = 1101101101102 = 66668)), and the only two known generalized Wieferich prime base 10 are 1685 and 5E685 (both of them end with 685 in dozenal (base 10)).
It is conjectured that there are infinitely many Wieferich primes in all bases, but this conjecture is not proven or disproven in any single base (except the trivial base 1, in which all odd primes are Wieferich primes).
base | known generalized Wieferich primes |
---|---|
2 | 771, 2047 (the Wieferich primes) |
3 | E, 406217 (the Mirimanoff primes) |
5 | 1002E, 1E51E, 15XX7E61, 39E028717, 1369315921, 306E7457755 |
7 | 5, 1E854E |
E | 5E |
11 | 5EE, 703407 |
15 | 3, 22771, 243XE, 7882491569E |
17 | 3, 7, 11, 37, E5, 19151E01 |
1E | 11, 9E8251, 470X031, 301X56181E, 35214X7XX155 |
base | known generalized Wieferich primes |
---|---|
4 | (exactly the Wieferich primes) |
6 | 32355, 21962E, 10804X5 |
8 | (exactly the Wieferich primes plus the prime 3) |
9 | (exactly the Mirimanoff primes) |
X | 3, 347, 16E557X1 |
10 | 1685, 5E685 |
Wieferich numbers[]
There are 88 known Wieferich numbers, they are
- 771, 1X93, 2047, 4517, 6119, 8281, X1XE, 11349, 16353, 20803, 224E7, 26589, 33X23, 46X39, 49687, 62009, 672X9, 77523, E009E, 124819, 179883, 1XX369, 290259, 372053, 4E5209, 830753, X96139, 12X3623, 1350957, 2091X39, 3X32449, 62356E3, 6513E3E, 8EE5631, E697123, 1485X307, 1733E9E9, 22EX4693, 2X849369, 38E93735, 42156919, 499EE5E3, 68E71839, 6E65332E, 88123X83, 98E4E941, E2E3X9X3, 106448353, 1255EX599, 161241377, 182X950E3, 18X739989, 252X2E403, 2989E8569, 317120X39, 3745E7553, 40890XX85, 463703XX9, 508843299, 5279E5523, 73868X009, 765E8661E, 8525E1483, 9493626E3, X68449151, 1022328813, 116X90E883, 1322109853, 137E5X4369, 19E1826023, 1X75E17659, 2137594209, 2423X67899, 2781123433, 3066982039, 348832E209, 3XEX570X83, 44959E9715, 5595076069, 57X594X753, 63XX540623, 7E0336X099, 91785060E3, X220989623, E8E7492809, 112455E4943, 145431X6183, 14E75427X39, 1E909X86253, 234E1316299, 26662524669, 337145X2409, 41409576509, 42XX407E6E3, 5E325816739, 6X293946853, 9X941567023, 1088701EX899, 1599750479E3, 18683E418139, 258404479069, 4554X311E599, 7510111E3183, 1X33033599509
There are only 7 known generalized Wieferich numbers base 10, and they are
- 1685, 5E685, 734X05, 95373E, 42E9X2E, 5746171E, 9362XXX1
Wieferich pairs[]
There are only 7 known Wieferich pairs, they are
- (2, 771), (3, 406217), (5, 39E028717), (5, 306E7457755), (6E, 299E), (63E, 134685), (181E, XX57)