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A123692 Primes p such that p^2 divides 5^(p-1) - 1.

%I A123692 #48 Jun 21 2025 11:49:19
%S A123692 2,20771,40487,53471161,1645333507,6692367337,188748146801
%N A123692 Primes p such that p^2 divides 5^(p-1) - 1.
%C A123692 Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
%C A123692 From _Felix Fröhlich_, Jan 06 2017: (Start)
%C A123692 a(6) and a(7) were found by Keller and Richstein (cf. Keller, Richstein, 2005).
%C A123692 Prime terms of A242959. (End)
%C A123692 From _Jianing Song_, Jun 21 2025: (Start)
%C A123692 The ring of integers of Q(5^(1/k)) is Z[5^(1/k)] if and only if k does not have a prime factor in this sequence (k is even or in A342391). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
%C A123692   (1 + sqrt(5))/2 is an algebraic integer, but it is not in Z[sqrt(5)];
%C A123692   (1 + 5^(10385/20771) + 5^(2*10385/20771) + ... + 5^(10384*10385/20771))/20771 is an algebraic integer, but it is not in Z[5^(1/20771)];
%C A123692   (1 + 5^(40486/40487) + 5^(2*40486/40487) + ... + 5^(40486*40486/40487))/40487 is an algebraic integer, but it is not in Z[5^(1/40487)]. (End)
%D A123692 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.
%H A123692 Amir Akbary and Sahar Siavashi, <a href="http://math.colgate.edu/~integers/s3/s3.Abstract.html">The Largest Known Wieferich Numbers</a>, INTEGERS, 18(2018), A3. See Table 1 p. 5.
%H A123692 Chris K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=FermatQuotient">Fermat quotient</a>.
%H A123692 Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf">The ring of integers in a radical extension</a>.
%H A123692 François G. Dorais and Dominic Klyve, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html">A Wieferich prime search up to p < 6.7*10^15</a>, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
%H A123692 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>, Math. Comp. 74 (2005), 927-936.
%H A123692 A. Paszkiewicz, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02090-5">A new prime p for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal</a>, Math. Comp. 78 (2009), 1193-1195.
%t A123692 Select[Prime[Range[2500]], Divisible[5^(# - 1) - 1, #^2] &] (* _Alonso del Arte_, Aug 01 2014 *)
%t A123692 Select[Prime[Range[55*10^6]],PowerMod[5,#-1,#^2]==1&] (* The program generates the first 4 terms of the sequence. *) (* _Harvey P. Dale_, Jan 29 2023 *)
%o A123692 (PARI)
%o A123692 N=10^9; default(primelimit, N);
%o A123692 forprime(n=2, N, if(Mod(5, n^2)^(n-1)==1, print1(n, ", ")));
%o A123692 \\ _Joerg Arndt_, May 01 2013
%Y A123692 Cf. A001220, A014127, A123693, A128667, A128668, A090968, A128669, A096082, A242959.
%K A123692 hard,nonn,more
%O A123692 1,1
%A A123692 _Max Alekseyev_, Oct 07 2006
%E A123692 More terms from _Alexander Adamchuk_, Nov 27 2006
%E A123692 Updated by _Max Alekseyev_, Jan 29 2012