A248173 Primes equal to 3 or congruent to 2 mod 3 that satisfy (1+a^p) == (1+a)^p (mod p^2) for all a between (p-3)/2.

3, 5, 11, 17, 23, 29, 41, 47, 53, 71, 89, 101, 107, 113, 131, 137, 149, 167, 173, 191, 197, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 431, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647

Offset

1,1

Comments

Apart from 3, subsequence of A003627.

Gives an easily testable condition which allows occasionally to prove the first case of Fermat’s Last Theorem over number fields for a prime number p == 2 mod 3.

Links

Table of n, a(n) for n=1..55.

Alain Kraus, Remarques sur le premier cas du théorème de Fermat sur les corps de nombres, arXiv:1410.0546 [math.NT], 2014, abstract in English.

Mathematica

selQ[p_] := p == 3 || Mod[p, 3] == 2 && AllTrue[Range[(p-3)/2], Mod[1+#^p, p^2] != Mod[(1+#)^p, p^2]&];
Select[Prime[Range[2, 120]], selQ] (* Jean-François Alcover, Sep 26 2018 *)

Prog

(PARI) isok(p) = {if ((p==3) || (p % 3) == 2, for (a=1, (p-3)/2, if (Mod(1+a^p, p^2) == Mod((1+a)^p, p^2), return (0)); ); return (1); ); return (0); }
lista(nn) = forprime(p=3, nn, if (isok(p), print1(p, ", ")));

Crossrefs

Cf. A003627.

Sequence in context: A216366 A136084 A045410 * A244029 A106902 A343048

Adjacent sequences: A248170 A248171 A248172 * A248174 A248175 A248176

Keyword

nonn

Author

Michel Marcus, Oct 03 2014

Status

approved