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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

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

Programs

Magma

[p: p in PrimesUpTo(1100) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 13 2012

Mathematica

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

PARI

is(n)=isprime(n) && ispower(Mod(2,n),8) \\ Charles R Greathouse IV, Feb 08 2017