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).

known generalized Wieferich primes in prime bases up to 20
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
known generalized Wieferich primes in composite bases up to 10
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)