A111027 Wieferich primes in base 12.

2693, 123653

Offset: 1

Author

Walter Kehowski, Oct 05 2005

Keywords

• nonn
• bref
• more

Comments

I have searched up to the 9 millionth prime, 160481183 and gave up trying to find a third term. The sequence is conjectured to be infinite. If the behavior is similar to base 10, A045616, then the next term could be greater than 2*10^11. In base 12 with X for ten and E for eleven the sequence is [1685, 5E685] so it would be interesting to see if the third term ends in 685 as well. These primes are also the Wieferich numbers in base 12: 12^phi(n) = 1 mod n^2.

Richard Fischer has carried this search to 4.8 * 10^13 (as of January 2014). - John Blythe Dobson, Mar 06 2014

Links

• Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
• Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2).

Crossrefs

Cf. A001220, A039951, A045616, A077815, A077816, A245529.

Programs

Maple

WP:=[]: for z from 1 to 1 do for k from 1 to 9000000 do p:=ithprime(k); if 12 &^(p-1) mod p^2 = 1 then WP:=[op(WP),p]; printf("p=%d, ",p); fi; if k mod 10^5 = 0 then printf("k=%d, ",k); fi; od; od; WP;

Mathematica

Select[Prime[Range[1000000]], PowerMod[12, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

Formula

12^(p-1) == 1 mod p^2