A049595 Primes p such that x^63 = 2 has a solution mod p.

2, 3, 5, 11, 17, 23, 31, 41, 47, 53, 59, 83, 89, 101, 107, 131, 137, 149, 157, 167, 173, 179, 191, 223, 227, 229, 233, 251, 257, 263, 269, 277, 283, 293, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 439, 443, 457, 461, 467, 479, 499, 503, 509, 521

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A059647 relative to A000040. - Vincenzo Librandi, Sep 15 2012

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• Index entries for related sequences

Crossrefs

Cf. A000040, A059647.

Programs

Magma

[p: p in PrimesUpTo(600) | exists(t){x : x in ResidueClassRing(p) | x^63 eq 2}]; // Vincenzo Librandi, Sep 15 2012

Maple

select(p -> isprime(p) and nops([msolve(x^63-2,p)])>0, [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Nov 03 2014

Mathematica

ok[p_]:= Reduce[Mod[x^63 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[150]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

PARI

N=10^4;
ok(p, r, k)={ return ( (p==r) || (Mod(r,p)^((p-1)/gcd(k,p-1))==1) ); }
forprime(p=2,N, if (ok(p,2,63),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

Python

from itertools import islice
from sympy import nextprime, is_nthpow_residue
def A049595_gen(startvalue=2): # generator of terms >= startvalue
    p = max(startvalue-1,1)
    while (p:=nextprime(p)):
        if is_nthpow_residue(2,63,p):
            yield p
A049595_list = list(islice(A049595_gen(),20)) # Chai Wah Wu, May 06 2024