A269627 Numbers n in S such that F(n) is prime: see comments for definitions.

9, 123, 3585, 16253, 17145, 79023, 100619, 501823, 696123

Offset

1,1

Comments

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.

Related to an efficient (quasiquadratic) deterministic primality test using elliptic curves over imaginary quadratic fields with complex multiplication.

Links

Table of n, a(n) for n=1..9.

Alexander Abatzoglou, Alice Silverberg, Andrew V. Sutherland, and Angela Wong, A framework for deterministic primality proving using elliptic curves with complex multiplication, Mathematics of Computation 85:299 (2016), pp. 1462-1483.

Example

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.

Prog

(PARI) F(k)=1-4*simplify(quadgen(-15)^k+(1-quadgen(-15))^k)+4^(k+2)
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", ")))

Crossrefs

Sequence in context: A295095 A295549 A211101 * A209504 A320529 A280896

Adjacent sequences: A269624 A269625 A269626 * A269628 A269629 A269630

Keyword

nonn,more

Author

Charles R Greathouse IV, Mar 01 2016

Status

approved