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A111027 Wieferich primes in base 12. 7
2693, 123653 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=1..2.

Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2).


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


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;


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

Sequence in context: A185748 A252404 A020424 * A245529 A210049 A115930

Adjacent sequences:  A111024 A111025 A111026 * A111028 A111029 A111030




Walter Kehowski, Oct 05 2005



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Last modified December 22 10:11 EST 2014. Contains 252339 sequences.