A040098 Primes p such that x^4 = 2 has a solution mod p.

2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 113, 127, 151, 167, 191, 199, 223, 233, 239, 257, 263, 271, 281, 311, 337, 353, 359, 367, 383, 431, 439, 463, 479, 487, 503, 577, 593, 599, 601, 607, 617, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

For a prime p congruent to 1 mod 8, 2 is a biquadratic residue mod p if and only if there are integers x,y such that x^2 + 64*y^2 = p. 2 is also a biquadratic residue mod 2 and mod p for any prime p congruent to 7 mod 8 and for no other primes. - Fred W. Helenius (fredh(AT)ix.netcom.com), Dec 30 2004

Complement of A040100 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Franz Lemmermeyer, Bibliography on Reciprocity Laws
• Index entries for related sequences

Crossrefs

Cf. A000040, A001132, A040028, A040100, A045315.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

Programs

Magma

[ p: p in PrimesUpTo(919) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 2} ]; // Klaus Brockhaus, Dec 02 2008

Mathematica

ok[p_] := Reduce[ Mod[x^4 - 2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200]], ok] (* Jean-François Alcover, Dec 14 2011 *)

PARI

forprime(p=2,2000,if([]~!=polrootsmod(x^4-2,p),print1(p,", ")));print(); \\ Joerg Arndt, Jul 27 2011