A003628 Primes congruent to {5, 7} mod 8. (Formerly M3764)

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

1,1

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

Formula

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

Mathematica

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

Prog

(Haskell)
a003628 n = a003628_list !! (n-1)
a003628_list = filter ((== 1) . a010051) a047566_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012
(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 */
(Magma) [ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // Vincenzo Librandi, Aug 22 2012

Crossrefs

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

Sequence in context: A216748 A216768 A216750 * A216776 A105874 A105904

Adjacent sequences: A003625 A003626 A003627 * A003629 A003630 A003631

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Mira Bernstein

Status

approved