A045315 Primes p such that x^8 = 2 has a solution mod p.

2, 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 257, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039

Offset

1,1

Comments

Coincides with the sequence of "primes p such that x^16 = 2 has a solution mod p" for first 58 terms (and then diverges).

Complement of A045316 relative to A000040. - Vincenzo Librandi, Sep 13 2012

References

A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und -2, Deutsche Math. 4 (1939), 44-52; FdM 65 - I (1939), 112.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

H. Hasse, Der 2^n-te Potenzcharakter von 2 im Koerper der 2^n-ten Einheitswurzeln, Rend. Circ. Matem. Palermo (2), 7 (1958), 185-243.

Franz Lemmermeyer, Bibliography on Reciprocity Laws

A. L. Whiteman, The sixteenth power residue character of 2, Canad. J. Math. 6 (1954), 364-373; Zbl 55.27102.

Index entries for related sequences

Mathematica

ok[p_] := Reduce[ Mod[x^8-2, p] == 0, x, Integers] =!= False; Select[ Prime[ Range[200] ], ok] (* Jean-François Alcover, Nov 28 2011 *)

Prog

(Magma) [p: p in PrimesUpTo(1100) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 13 2012
(PARI) is(n)=isprime(n) && ispower(Mod(2, n), 8) \\ Charles R Greathouse IV, Feb 08 2017

Crossrefs

Cf. A000040, A001132, A040028, A040098, A045316.

Sequence in context: A309580 A186098 A040098 * A072935 A049564 A072936

Adjacent sequences: A045312 A045313 A045314 * A045316 A045317 A045318

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved