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Primes p such that x^12 = 2 has no solution mod p.
+10
7
3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 79, 83, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 199, 211, 227, 229, 241, 251, 269, 271, 277, 283, 293, 307, 313, 317, 331, 337, 347, 349, 367, 373, 379, 389
COMMENTS
Coincides for the first 119 terms with sequence of primes p such that x^36 = 2 has no solution mod p (first divergence is at 919, cf. A059668).
MATHEMATICA
ok[p_] := Reduce[Mod[x^12 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 14 2012 *)
Primes p such that x^36 = 2 has no solution mod p, but x^12 = 2 has a solution mod p.
+10
4
919, 1423, 1999, 2143, 2287, 2791, 4177, 4519, 4663, 5113, 5167, 6679, 6967, 8713, 9631, 9649, 9721, 11863, 12241, 12583, 12799, 13591, 16111, 17551, 18127, 20359, 20719, 21529, 21727, 21799, 22807, 23041, 23473, 23743, 23833, 23887, 23977
MATHEMATICA
Select[Prime[Range[PrimePi[30000]]], ! MemberQ[PowerMod[Range[#], 36, #], Mod[2, #]]&& MemberQ[PowerMod[Range[#], 12, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 22 2013 *)
PROG
(Magma) [p: p in PrimesUpTo(24000) | not exists{x: x in ResidueClassRing(p) | x^36 eq 2} and exists{x: x in ResidueClassRing(p) | x^12 eq 2}]; // Vincenzo Librandi, Sep 21 2012
Primes p such that x^6 = 2 has a solution mod p, but x^(6^2) = 2 has no solution mod p.
+10
3
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
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, ", ")));
(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)
CROSSREFS
Cf. A040992, A049568, A059264, A059667, A070179, A070180, A070181, A070182, A070184, A070185, A070186, A070187, A070188.
Primes p such that x^36 = 2 has no solution mod p.
+10
1
3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 79, 83, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 199, 211, 227, 229, 241, 251, 269, 271, 277, 283, 293, 307, 313, 317, 331, 337, 347, 349, 367, 373, 379, 389
COMMENTS
Coincides for the first 416 terms with the sequence of primes p such that x^108 = 2 has no solution mod p (first divergence is at 3947). [ Bruno Berselli, Sep 14 2012]
MATHEMATICA
ok[p_] := Reduce[Mod[x^36 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[500]], ok]
Select[Prime[Range[PrimePi[400]]], ! MemberQ[PowerMod[Range[#], 36, #], Mod[2, #]] &] (* Bruno Berselli, Sep 14 2012 *)
PROG
(Magma) [p: p in PrimesUpTo(400) | forall{x: x in ResidueClassRing(p) | x^36 ne 2}];
CROSSREFS
Cf. A059668 (primes p such that x^36 = 2 has no solution mod p, but x^12 = 2 has a solution mod p).
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