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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)).

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 Edit

There are 88 known Wieferich numbers, the Wieferich numbers below 106 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

There are only 7 known generalized Wieferich numbers base 10, and they are

1685, 5E685, 734X05, 95373E, 42E9X2E, 5746171E, 9362XXX1

Wieferich pairs Edit

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