A342391 -id:A342391 - cp's OEIS frontend

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

2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

Offset: 1

Max Alekseyev, Oct 07 2006

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
From Felix Fröhlich, Jan 06 2017: (Start)
a(6) and a(7) were found by Keller and Richstein (cf. Keller, Richstein, 2005).
Prime terms of A242959. (End)
From Jianing Song, Jun 21 2025: (Start)
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:
  (1 + sqrt(5))/2 is an algebraic integer, but it is not in Z[sqrt(5)];
  (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)];
  (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)

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Chris K. Caldwell, The Prime Glossary, Fermat quotient.
Keith Conrad, The ring of integers in a radical extension.
François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
W. Keller and J. Richstein, Solutions of the congruence a^p-1 == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
A. Paszkiewicz, A new prime p for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.

Cf. A001220, A014127, A123693, A128667, A128668, A090968, A128669, A096082, A242959.

Select[Prime[Range[2500]], Divisible[5^(# - 1) - 1, #^2] &] (* Alonso del Arte, Aug 01 2014 *)
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 *)

N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(5, n^2)^(n-1)==1, print1(n, ", ")));
\\ Joerg Arndt, May 01 2013

More terms from Alexander Adamchuk, Nov 27 2006

Updated by Max Alekseyev, Jan 29 2012

A342390 Numbers k such that the ring of integers of Q(2^(1/k)) is not Z[2^(1/k)].

Original entry on oeis.org

1093, 2186, 3279, 3511, 4372, 5465, 6558, 7022, 7651, 8744, 9837, 10533, 10930, 12023, 13116, 14044, 14209, 15302, 16395, 17488, 17555, 18581, 19674, 20767, 21066, 21860, 22953, 24046, 24577, 25139, 26232, 27325, 28088, 28418, 29511, 30604, 31599, 31697, 32790

Offset: 1

Jianing Song, Mar 10 2021

nonn

For k > 1, a != 1 being a squarefree number (a != -1 unless k is a power of 2), then the ring of integers of Q(a^(1/k)) is Z[a^(1/k)] if and only if: for every p dividing k, we have a^(p-1) !== 1 (mod p^2). In other words, O_Q(a^(1/k)) = Z[a^(1/k)] if and only if none of the prime factors of k is a Wieferich prime of base a. See Theorem 5.3 of the paper of Keith Conrad.
In general, if a^d == 1 (mod p^2) for some d|(p-1), then it is easy to show that x = (1 + a^(d/p) + a^(2*d/p) + ... + a^((p-1)*d/p))/p is an algebraic integer not in Z[a^(1/p)].
Here a = 2, and the only known Wieferich primes of base 2 (A001220) are 1093, 3511 are no more below 4.97*10^17. So all known terms are multiples of either 1093 or 3511 (or both).

			2^364 == 1 (mod 1093^2), so x = (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer not in Z[2^(1/1093)].

Jianing Song, Table of n, a(n) for n = 1..11995 (all terms up to 10^7).
Keith Conrad, The ring of integers in a radical extension

Cf. A001220, A342391.

Wieferich_up_to_n(lim) = my(v=[]); forprime(p=2, lim, if(Mod(2,p^2)^(p-1)==1, v=concat(v,p))); v
Up_to_n(lim) = my(pv=Wieferich_up_to_n(lim), list=[]); for(i=1, #pv, my(p=pv[i]); for(k=1, lim\p, list=concat(list, k*p))); list=Set(list); list \\ corrected by Jianing Song, Mar 19 2022

Up_to_n(lim) = my(list=[]); for(k=1, lim\1093, list=concat(list, 1093*k)); for(k=1, lim\3511, list=concat(list, 3511*k)); list=Set(list); list \\ valid up to 4.97*10^17, corrected by Jianing Song, Mar 19 2022

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

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A342390 Numbers k such that the ring of integers of Q(2^(1/k)) is not Z[2^(1/k)].

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