A003628 Primes congruent to {5, 7} mod 8.

5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 263, 269, 271, 277, 293, 311, 317, 349, 359, 367, 373, 383, 389, 397, 421, 431, 439

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

Inert rational odd primes in the field Q(sqrt(-2)).

Primes p such that p XOR 5 = p - 5. - Brad Clardy, Jul 22 2012

Terms m in A047566 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

This sequence gives the primes p which satisfy norm(rho(p)) = - 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For p == 5 (mod 8) the norm is C(p, 0) (see a comment on 2*A230076) and for p == 7 (mod 8) the norm is -C(p, 0) (see a comment on A186302). For the primes with norm(rho(p)) = +1 see A033200. - Wolfdieter Lang, Oct 24 2013

References

• H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A000040, A039706, A033203 (complement with respect to A000040).

Programs

Haskell

a003628 n = a003628_list !! (n-1)
a003628_list = filter ((== 1) . a010051) a047566_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

Magma

[ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // Vincenzo Librandi, Aug 22 2012

Mathematica

Select[Prime[Range[200]],MemberQ[{5,7},Mod[#,8]]&] (* Harvey P. Dale, Oct 24 2011 *)

PARI

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -2, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 24 2023