A070183 Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.

17, 41, 137, 401, 433, 449, 457, 521, 569, 641, 761, 809, 857, 919, 929, 953, 977, 1361, 1409, 1423, 1657, 1697, 1999, 2017, 2081, 2143, 2153, 2287, 2297, 2417, 2609, 2633, 2729, 2753, 2777, 2791, 2801, 2897, 2953, 3041, 3209, 3329, 3457, 3593, 3617

Offset

1,1

Links

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

Maple

select(p -> isprime(p) and [msolve(x^6=2, p)]<>[] and [msolve(x^36=2, p)]=[] , [seq(i, i=3..10^4, 2)]); # Robert Israel, May 13 2018

Prog

(PARI) forprime(p=2, 3700, x=0; while(x<p&&x^6%p!=2%p, x++); if(x<p, y=0; while(y<p&&y^(6^2)%p!=2%p, y++); if(y==p, print1(p, ", "))))
(Magma) [p: p in PrimesUpTo(5000) | not exists{x: x in ResidueClassRing(p) | x^36 eq 2} and exists{x: x in ResidueClassRing(p) | x^6 eq 2}]; // Vincenzo Librandi, Sep 21 2012
(PARI)
ok(p, r, k1, k2)={
    if (  Mod(r, p)^((p-1)/gcd(k1, p-1))!=1, return(0) );
    if (  Mod(r, p)^((p-1)/gcd(k2, p-1))==1, return(0) );
    return(1);
}
forprime(p=2, 10^4, if (ok(p, 2, 6, 6^2), print1(p, ", ")));
/* Joerg Arndt, Sep 21 2012 */
(Python)
from itertools import count, islice
from sympy import nextprime, is_nthpow_residue
def A070183_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1), 2)
    while True:
        if is_nthpow_residue(2, 6, p) and not is_nthpow_residue(2, 36, p):
            yield p
        p = nextprime(p)
A070183_list = list(islice(A070183_gen(), 20)) # Chai Wah Wu, May 02 2024

Crossrefs

Cf. A040992, A049568, A059264, A059667, A070179, A070180, A070181, A070182, A070184, A070185, A070186, A070187, A070188.

Sequence in context: A139961 A188661 A205171 * A201705 A165668 A269425

Adjacent sequences: A070180 A070181 A070182 * A070184 A070185 A070186

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 29 2002

Status

approved