

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 (p3)/2.


0



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
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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[(p3)/2], Mod[1+#^p, p^2] != Mod[(1+#)^p, p^2]&];
Select[Prime[Range[2, 120]], selQ] (* JeanFrançois Alcover, Sep 26 2018 *)


PROG

(PARI) isok(p) = {if ((p==3)  (p % 3) == 2, for (a=1, (p3)/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 A293711
Adjacent sequences: A248170 A248171 A248172 * A248174 A248175 A248176


KEYWORD

nonn


AUTHOR

Michel Marcus, Oct 03 2014


STATUS

approved



