A059287 Primes p such that x^16 = 2 has no solution mod p, but x^8 = 2 has a solution mod p.

1217, 1249, 1553, 1777, 2833, 4049, 4273, 4481, 4993, 5297, 6449, 6481, 6689, 7121, 8081, 8609, 9137, 9281, 9649, 10337, 10369, 10433, 11329, 11617, 11633, 12241, 12577, 13121, 13441, 13633, 14321, 14753, 15121, 15569, 16417, 16433, 16673

Offset: 1

Author

Klaus Brockhaus, Jan 25 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A000040, A045315, A045316.

Cf. A070184 (same with x^64 instead of x^16).

Programs

Magma

[p: p in PrimesUpTo(17000) | not exists{x: x in ResidueClassRing(p) | x^16 eq 2} and exists{x: x in ResidueClassRing(p) | x^8 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Mathematica

Select[Prime[Range[PrimePi[20000]]], !MemberQ[PowerMod[Range[#], 16, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 8, #], Mod[2, #]]&] (* Vincenzo Librandi, Sep 21 2013 *)

PARI

select( {is_A059287(p)=Mod(2,p)^(p\gcd(8,p-1))==1&&Mod(2,p)^(p\gcd(16,p-1))!=1}, primes(1999)) \\ Could any composite number pass this test? - M. F. Hasler, Jun 22 2024

Python

from itertools import islice
from sympy import is_nthpow_residue, nextprime
def A059287_gen(startvalue=2): # generator of terms >= startvalue
    p = max(1,startvalue-1)
    while (p:=nextprime(p)):
        if is_nthpow_residue(2,8,p) and not is_nthpow_residue(2,16,p):
            yield p
A059287_list = list(islice(A059287_gen(),10)) # Chai Wah Wu, Jun 23 2024