A222206 Least prime p such that q^(p-1) == 1 (mod p^2) for n primes q < p.

2, 11, 349, 13691, 24329

Offset: 0

Author

Jonathan Sondow, Feb 12 2013

Keywords

• nonn
• more

Comments

I found no new terms < 5*10^6. - J. Stauduhar, Mar 23 2013

a(5) > 13*10^6, if it exists. Note that, up to 13*10^6, the only other prime p (apart 24329) such that the congruence is satisfied for 4 primes q < p is 9656869. - Giovanni Resta, May 23 2017

Examples

			For the prime p = 349, but for no smaller prime, there are 2 primes q = 223 and 317 < p with  q^(p-1) == 1 (mod p^2), so a(2) = 349.
		

References

• L. E. Dickson, History of the Theory of Numbers, vol. 1, chap. IV.

Links

• W. Keller and J. Richstein, Fermat quotients that are divisible by p. [Broken link]
• Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod (p^r)), Math. Comp. 74 (2005), 927-936.

Crossrefs

Cf. A001220, A039678, A134307, A143548, A222184, A222185.

Programs

Mathematica

f[n_] := Block[{p = 2, q = {}}, While[ Count[ PowerMod[ q, p - 1, p^2], 1] != n, q = Join[q, {p}]; p = NextPrime@ p]; p]; Array[f, 5, 0] (* Robert G. Wilson v, Mar 09 2015 *)

PARI

a(n) = {nb = 0; p = 2; while (nb != n, p = nextprime(p+1); nb = 0; forprime(q=2, p-1, if (Mod(q, p^2)^(p-1) == 1, nb ++); if (nb > n, break););); p;} \\ Michel Marcus, Mar 08 2015