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%I #41 Mar 25 2018 08:21:25
%S 2,1093,83,4871,2903,18787,911,318917,3,1006003,5,1645333507
%N Two-column array A(n, k) read by rows, where A(n, 1) and A(n, 2) respectively give values of q and p in the n-th double Wieferich prime pair, where p > q. Terms sorted first by increasing size of p, then by increasing size of q.
%C Double Wieferich prime pairs are pairs of prime numbers p and q such that p^(q-1) == 1 (mod q^2) and q^(p-1) == 1 (mod p^2).
%C Pairs of primes p and q such that A274916(x, y) = 2, where x and y are the indices of p and q in A000040 respectively.
%C This sequence provides a "complete" listing of double Wieferich prime pairs. A124122 omits values of p where a smaller p with the same value of q in A124121 exists, while A266829 lists each value of p exactly once, regardless of how many values of q exist for that p.
%C There are two ways to retrieve the pair (p, q) from the data when looking at any specific value:
%C 1. Look at the indices i of the terms. If the index is even, a(i) is the larger member p of a pair, so q is a(i-1). If the index is odd, a(i) is the smaller member q of the pair, so p is a(i+1).
%C 2. Look at a term a(i) in the data section. If a(i-1) and a(i+1) are both larger than a(i), then a(i) is the smaller member of a pair and its partner is a(i+1). If a(i-1) and a(i+1) are both smaller than a(i), then a(i) is the larger member of a pair and its partner is a(i-1).
%C There are no further pairs with p*q <= 10^15 and p < 2^(1/3)*10^10 and only one additional Wieferich pair, namely (5, 188748146801), is known, but its position in the sequence is uncertain (cf. Logan, Mossinghoff, 2015).
%C Double Wieferich pairs were first mentioned by Inkeri, who showed that if Catalan's Diophantine equation x^p - y^q = 1 has a solution (x, y) for prime numbers p and q both congruent to 3 modulo 4, p > q and h(p) =/= 0 (mod q), where h(p) is the class number of the field k(sqrt(-p)), then (p, q) is a double Wieferich pair (cf. Inkeri, 1964, Theorem 2).
%C The pairs (2, 1093) and (83, 4871) were apparently first found by Aaltonen and Inkeri, who state that these two and a third one, (3, 1006003), from a table from Brillhart, Tonascia and Weinberger, are the only pairs they are aware of (cf. Aaltonen, Inkeri, 1991, Remark on p. 365).
%C The pairs (2903, 18787) and (911, 318917) were first found by Mignotte and Roy (cf. Keller, Richstein, 2005, p. 935) and later also by Ernvall and Metsänkylä in a search to 10^6, who also mention the pair (5, 1645333507) found by Montgomery (cf. Ernvall, Metsänkylä, 1997, p. 1360).
%C Several further conditions connecting double Wieferich pairs to Catalan's equation were obtained by Steiner, for example the result that if both p and q in a solution to the Catalan equation are congruent to 3 modulo 4 or both p and q satisfy either p == 3 (mod 4) and q == 5 (mod 8) or vice versa, then (p, q) is a double Wieferich pair (cf. Steiner, 1998, Theorems 7 and 8).
%C Mihailescu further showed that if the Catalan equation has a solution with p and q distinct odd primes and xy != 0, then q^2 | x, p^2 | y and p and q form a double Wieferich prime pair (cf. Mihailescu, 2003, Theorem 1).
%C If the Diophantine equation p^x - q^y = c has more than one solution with q an odd prime incongruent to 1 modulo 12, p < 2*q, gcd(p-1, q-1) even and (p, q, c) not one of (3, 2, 1), (2, 3, 5), (2, 3, 13), (2, 5, 3) or (13, 3, 10), then (p, q) is a double Wieferich pair (cf. Scott, Styer, 2004, pages 218-219).
%C Corollary to Proposition 4 in Pomerance, Selfridge, Wagstaff, 1980: Let p and q be primes and let c and d be composites such that c and d are base-p and base-q Fermat pseudoprimes, respectively. If p^2 divides d and q^2 divides c, then (p, q) is a double Wieferich pair.
%H M. Aaltonen and K. Inkeri, <a href="https://doi.org/10.1090/S0025-5718-1991-1052082-7">Catalan's equation x^p - y^p = 1 and related congruences</a>, Math. Comp. 56 (1991), 359-370.
%H R. Ernvall and T. Metsänkylä, <a href="https://doi.org/10.1090/S0025-5718-97-00843-0">On the p-divisibility of Fermat quotients</a>, Mathematics of Computation 66 (1997), 1353-1365.
%H K. Inkeri, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/9/3/75477/on-catalan-apos-s-problem">On Catalan's problem</a>, Acta Arithmetica 9 (1964), 285-290.
%H W. Keller and J. Richstein, <a href="https://doi.org/10.1090/S0025-5718-04-01666-7">Solutions of the congruence a^p-1 == 1 (mod p^r)</a>, Mathematics of Computation, 74 (2005), 927-936.
%H B. Logan and M. J. Mossinghoff, <a href="https://www.researchgate.net/publication/281628524">Double Wieferich pairs and circulant Hadamard matrices</a>, ResearchGate, 2015.
%H P. Mihailescu, <a href="https://doi.org/10.1016/S0022-314X(02)00101-4">A class number free criterion for catalan's conjecture</a>, Journal of Number Theory, Vol. 99, No. 2 (2003), 225-231.
%H C. Pomerance, J. L. Selfridge and S. S. Wagstaff, <a href="https://doi.org/10.1090/S0025-5718-1980-0572872-7">The pseudoprimes to 25 * 10^9</a>, Mathematics of Computation, 35 (1980), 1003-1026.
%H R. Scott and R. Styer, <a href="https://doi.org/10.1016/j.jnt.2003.11.008">On p^x-q^y=c and related three term exponential Diophantine equations with prime bases</a>, Journal of Number Theory, Vol. 105, No. 2 (2004), 212-234.
%H R. Steiner, <a href="https://doi.org/10.1090/S0025-5718-98-00966-1">Class number bounds and Catalan's equation</a>, Mathematics of Computation 67 (1998), 1317-1322.
%e The primes 83 and 4871 satisfy 83^4870 == 1 (mod 23726641) and 4871^82 == 1 (mod 6889), respectively, so 83 and 4871 are terms of the sequence.
%o (PARI) is_dwpp(n, k) = Mod(n, k^2)^(k-1)==1 && Mod(k, n^2)^(n-1)==1
%o search(x, y) = forprime(p=x, y, forprime(q=1, p-1, if(is_dwpp(p, q), print1(q, ", ", p, ", "))))
%o search(1, 1e6) \\ search pairs in the interval [1, 10^6]
%Y Cf. A000040, A124121, A124122, A266829, A274916.
%K nonn,tabf,hard,more
%O 1,1
%A _Felix Fröhlich_, Feb 11 2017
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