%I #11 Sep 08 2022 08:45:05

%S 113,281,353,593,617,919,1049,1097,1193,1217,1423,1481,1553,1601,1753,

%T 1777,1889,1999,2129,2143,2273,2281,2287,2393,2689,2791,2833,3089,

%U 3137,3761,3833,4001,4049,4153,4177,4217,4289,4457,4481,4519,4657,4663,4817

%N Primes p such that x^12 = 2 has a solution mod p, but x^(12^2) = 2 has no solution mod p.

%o (PARI) forprime(p=2,5000,x=0; while(x<p&&x^12%p!=2%p,x++); if(x<p,y=0; while(y<p&&y^(12^2)%p!=2%p,y++); if(y==p,print1(p,","))))

%o (Magma) [p: p in PrimesUpTo(5000) | not exists{x: x in ResidueClassRing(p) | x^144 eq 2} and exists{x: x in ResidueClassRing(p) | x^12 eq 2}]; // _Vincenzo Librandi_, Sep 21 2012

%o (PARI)

%o ok(p, r, k1, k2)={

%o if ( Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );

%o if ( Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );

%o return(1);

%o }

%o forprime(p=2,10^5, if (ok(p,2,12,12^2),print1(p,", ")));

%o /* _Joerg Arndt_, Sep 21 2012 */

%Y Cf. A049544, A059667, A070179 - A070187.

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 29 2002