A040098 - OEIS

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

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OFFSET

1,1

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

MATHEMATICA

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

PROG

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

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

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, ...

Sequence in context: A042145 A309580 A186098 * A045315 A072935 A049564

Adjacent sequences: A040095 A040096 A040097 * A040099 A040100 A040101

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved