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%I #16 Feb 11 2024 17:27:49
%S 13,28393,128758492789,73,193,37,457,8363172060732903211423577787181
%N Primes of the form 12k+1 generated recursively. Initial prime is 13. General term is a(n)=Min {p is prime; p divides Q^4-Q^2+1}, where Q is the product of previous terms in the sequence.
%C All prime divisors of Q^4 - Q^2 + 1 are congruent to 1 modulo 12.
%D K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, Second Edition (1990), p. 63.
%e a(3) = 128758492789 is the smallest prime divisor of Q^4 - Q^2 + 1 = 18561733755472408508281 = 128758492789 * 144159296629, where Q = 13 * 28393.
%t a = {13}; q = 1;
%t For[n = 2, n ≤ 8, n++,
%t q = q*Last[a];
%t AppendTo[a, Min[Select[FactorInteger[q^4 - q^2 + 1][[All, 1]],
%t Mod[#, 12] == 1 &]]];
%t ];
%t a (* _Robert Price_, Jun 25 2015 *)
%Y Cf. A000945, A068228, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.
%K more,nonn
%O 1,1
%A _Nick Hobson_, Nov 18 2006
%E a(8) from _Robert Price_, Jun 25 2015
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