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%I #9 Mar 02 2016 09:17:21
%S 9,123,3585,16253,17145,79023,100619,501823,696123
%N Numbers n in S such that F(n) is prime: see comments for definitions.
%C Define S = {9, 19, 39, 45, 59, 63, 67, 85, 105, 123, 129, 133, 159, 169, 173, 181, 183, 221, 223, 225, 229 mod 240} and F(n) = 1 - 4(alpha^n + conj(alpha)^n) + 4^(n+2), where alpha = (1 + sqrt(-15))/2.
%C Related to an efficient (quasiquadratic) deterministic primality test using elliptic curves over imaginary quadratic fields with complex multiplication.
%H Alexander Abatzoglou, Alice Silverberg, Andrew V. Sutherland, and Angela Wong, <a href="http://arxiv.org/abs/1404.0107">A framework for deterministic primality proving using elliptic curves with complex multiplication</a>, Mathematics of Computation 85:299 (2016), pp. 1462-1483.
%e With alpha = (1 + sqrt(-15))/2, F(9) = 1 - 4(alpha^9 + conj(alpha)^9) + 4^11 = 1 - 4(((1 + sqrt(-15))/2)^9 + ((1 - sqrt(-15))/2)^9) + 4^11 = 1 - 4*781 + 4194304 = 4191181, which is prime. Since 9 is also in S, 9 is a member of the sequence.
%o (PARI) F(k)=1-4*simplify(quadgen(-15)^k+(1-quadgen(-15))^k)+4^(k+2)
%o forstep(n=9,1e5, [10, 20, 6, 14, 4, 4, 18, 20, 18, 6, 4, 26, 10, 4, 8, 2, 38, 2, 2, 4, 20], if(ispseudoprime(F(n)), print1(n", ")))
%K nonn,more
%O 1,1
%A _Charles R Greathouse IV_, Mar 01 2016