A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.

3, 11, 1091, 1296216011, 2177870960662059587828905091, 76870667, 19, 257680660619, 73677606898727076965233531, 23842300525435506904690028531941969449780447746432390747, 35164737203

Offset: 1

Author

Nick Hobson, Nov 18 2006

Keywords

• nonn

Comments

2+Q^2 always has a prime divisor congruent to 3 modulo 8.

Examples

			a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.
		

References

• D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Links

• Robert Price, Table of n, a(n) for n = 1..15
• N. Hobson, Home page (listed in lieu of email address)

Crossrefs

Cf. A000945, A007520, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

Mathematica

a = {3}; q = 1;
For[n = 2, n ≤ 5, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[2 + q^2][[All, 1]], Mod[#,
    8] \[Equal] 3 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

PARI

lista(nn) = my(f, q=3); print1(q); for(n=2, nn, f=factor(2+q^2)[, 1]~; for(i=1, #f, if(f[i]%8==3, print1(", ", f[i]); q*=f[i]; break))); \\ Jinyuan Wang, Aug 05 2022

Extensions

a(10) from Robert Price, Jul 04 2015

a(11) from Robert Price, Jul 05 2015