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%I #7 Sep 26 2018 11:26:59
%S 3,5,11,17,23,29,41,47,53,71,89,101,107,113,131,137,149,167,173,191,
%T 197,233,239,251,257,263,269,281,293,311,317,347,353,359,383,389,401,
%U 431,449,461,467,479,491,503,509,521,557,563,569,587,593,599,617,641,647
%N 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.
%C Apart from 3, subsequence of A003627.
%C 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.
%H Alain Kraus, <a href="http://arxiv.org/abs/1410.0546">Remarques sur le premier cas du théorème de Fermat sur les corps de nombres</a>, arXiv:1410.0546 [math.NT], 2014, abstract in English.
%t selQ[p_] := p == 3 || Mod[p, 3] == 2 && AllTrue[Range[(p-3)/2], Mod[1+#^p, p^2] != Mod[(1+#)^p, p^2]&];
%t Select[Prime[Range[2, 120]], selQ] (* _Jean-François Alcover_, Sep 26 2018 *)
%o (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);}
%o lista(nn) = forprime(p=3, nn, if (isok(p), print1(p, ", ")));
%Y Cf. A003627.
%K nonn
%O 1,1
%A _Michel Marcus_, Oct 03 2014
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