A124990 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.

13, 28393, 128758492789, 73, 193, 37, 457, 8363172060732903211423577787181

Offset: 1

Author

Nick Hobson, Nov 18 2006

Keywords

• more
• nonn

Comments

All prime divisors of Q^4 - Q^2 + 1 are congruent to 1 modulo 12.

Examples

			a(3) = 128758492789 is the smallest prime divisor of Q^4 - Q^2 + 1 = 18561733755472408508281 = 128758492789 * 144159296629, where Q = 13 * 28393.
		

References

• K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, Second Edition (1990), p. 63.

Crossrefs

Cf. A000945, A068228, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

Mathematica

a = {13}; q = 1;
For[n = 2, n ≤ 8, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[q^4 - q^2 + 1][[All, 1]],
    Mod[#, 12] == 1 &]]];
    ];
a  (* Robert Price, Jun 25 2015 *)

Extensions

a(8) from Robert Price, Jun 25 2015