cp's OEIS Frontend

A124121 Primes q which are the lesser of a double Wieferich prime pair. (List of known values of q in increasing order, without multiplicity.)

%I A124121 #26 Feb 16 2025 08:33:03
%S A124121 2,3,5,83,911,2903
%N A124121 Primes q which are the lesser of a double Wieferich prime pair. (List of known values of q in increasing order, without multiplicity.)
%C A124121 Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2). This sequence gives the primes q which are the lesser member (listed second) of such pairs, in increasing order, but without multiplicity:
%C A124121 For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). In this sequence, 5 is only listed once, as a(3), and only the lesser value p=1645333507 is listed as A124122(3).
%C A124121 This is just the list of known pairs: there may be gaps, i.e. missing primes.
%H A124121 Yuri F. Bilu, <a href="https://eudml.org/doc/252135">Catalan's Conjecture</a>, Seminaire Bourbaki, (2002-2003).
%H A124121 Michael Mossinghoff, <a href="http://www.cecm.sfu.ca/~mjm/WieferichBarker">Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices</a>, as of Feb 12 2009.
%H A124121 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleWieferichPrimePair.html">Double Wieferich Prime Pair</a>
%H A124121 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_pair">Wieferich pair</a>
%o A124121 (PARI) /* The following (highly unoptimized) code misses the value q=5 (corresponding to a very large value of p) */
%o A124121 default(primelimit,1010000); forprime(q=1, default(primelimit), forprime(p=q+1, default(primelimit),  Mod(p, q^2)^(q-1)-1 & next; Mod(q, p^2)^(p-1)-1 || print1( q", ") || break))  \\ _M. F. Hasler_, Oct 08 2011
%Y A124121 See A124122 for values of p.
%Y A124121 Cf. A196511, A196733.
%K A124121 nonn,hard,more
%O A124121 1,1
%A A124121 _N. J. A. Sloane_, following an email from _Robert G. Wilson v_, Nov 30 2006