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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.
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
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
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
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 03 2014
STATUS
approved