A014755 3 and -3 are both 4th powers (one implies other) mod these primes p=1 mod 8.

193, 313, 433, 577, 601, 673, 769, 937, 1201, 1297, 1321, 1657, 1801, 1993, 2137, 2473, 2521, 2593, 2833, 2953, 3169, 3529, 3673, 3697, 3769, 3889, 4057, 4129, 4153, 4297, 4441, 4513, 4561, 4801, 4969, 5113, 5209, 5233, 5281, 5449, 5521

Offset: 1

Author

Warren D. Smith

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007519.

Programs

Maple

filter:= proc(p) isprime(p) and [msolve(x^4=3, p)] <> [] end proc:
select(filter, [seq(i,i=1..10^4, 8)]); # Robert Israel, May 07 2019

Mathematica

okQ[p_] := PrimeQ[p] && Solve[x^4 == 3, x, Modulus -> p] != {};
Select[Range[1, 10000, 8], okQ] (* Jean-François Alcover, Feb 08 2023 *)

PARI

forprime(p=1,9999,p%8==1&&ispower(Mod(3,p),4)&&print1(p",")) \\ M. F. Hasler, Feb 18 2014

PARI

is_A014755(p)={p%8==1&&ispower(Mod(3,p),4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Python

from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A014755_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        if p&7==1 and is_nthpow_residue(3,4,p) and is_nthpow_residue(-3,4,p):
            yield p
        p = nextprime(p)
A014755_list = list(islice(A014755_gen(),20)) # Chai Wah Wu, May 02 2024

Extensions

Offset changed from 0 to 1 by Bruno Berselli, Feb 20 2014