login
A070186
Primes p such that x^10 = 2 has a solution mod p, but x^(10^2) = 2 has no solution mod p.
4
17, 97, 137, 151, 193, 241, 313, 409, 433, 449, 457, 569, 641, 673, 769, 809, 857, 929, 953, 977, 1009, 1129, 1297, 1409, 1489, 1657, 1697, 1873, 1993, 2017, 2137, 2153, 2297, 2377, 2417, 2609, 2617, 2633, 2713, 2729, 2753, 2777, 2897, 2953, 3169, 3209
OFFSET
1,1
PROG
(PARI) forprime(p=2, 3300, x=0; while(x<p&&x^10%p!=2%p, x++); if(x<p, y=0; while(y<p&&y^(10^2)%p!=2%p, y++); if(y==p, print1(p, ", "))))
(Magma) [p: p in PrimesUpTo(3500) | not exists{x: x in ResidueClassRing(p) | x^100 eq 2} and exists{x: x in ResidueClassRing(p) | x^10 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, 10^5, if (ok(p, 2, 10, 10^2), print1(p, ", "))); \\ A070186
/* Joerg Arndt, Sep 21 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 29 2002
STATUS
approved