A059262 -id:A059262 - cp's OEIS frontend

A049596 Primes p such that x^9 = 2 has a solution mod p.

2, 3, 5, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 71, 83, 89, 101, 107, 113, 127, 131, 137, 149, 157, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 439, 443, 449

Offset: 1

N. J. A. Sloane

Coincides with sequence of "primes p such that x^27 = 2 has a solution mod p" for first 339 terms, then diverges.

Complement of A059262 relative to A000040. - Vincenzo Librandi, Sep 15 2012

			0^9 == 2 (mod 2). 2^9 == 2 (mod 3). 2^9 == 2 (mod 5). 6^9 == 2 (mod 11). 2^9 == 2 (mod 17). 9^9 == 2 (mod 23). 11^9 == 2 (mod 29). 16^9 == 2 (mod 31). 20^9 == 2 (mod 41). 26^9 == 2 (mod 43). - _R. J. Mathar_, Jul 20 2025

R. J. Mathar, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A001132, A059262.

[p: p in PrimesUpTo(500) | exists(t){x : x in ResidueClassRing(p) | x^9 eq 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^9 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

A059354 Primes p such that x^27 = 2 has no solution mod p, but x^9 = 2 has a solution mod p.

Original entry on oeis.org

3943, 11287, 12853, 14149, 17659, 20143, 21061, 21277, 23059, 23599, 25759, 26407, 26731, 29863, 32833, 33751, 35803, 37747, 38287, 39367, 39799, 46441, 47737, 47791, 57781, 59887, 61291, 62047, 63127, 65557, 68311, 71443, 73063, 78301

Offset: 1

Klaus Brockhaus, Jan 27 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..107

Cf. A000040, A049596, A059262, A070185.

[ p: p in PrimesUpTo(80000) | exists(t){x: x in ResidueClassRing(p) | x^9 eq 2} and forall(t){x : x in ResidueClassRing(p) | x^27 ne 2} ]; // Klaus Brockhaus, Dec 05 2008

Select[Prime[Range[PrimePi[80000]]], !MemberQ[PowerMod[Range[#], 27, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 9, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 21 2013 *)

a(25)-a(34) from Klaus Brockhaus, Dec 05 2008

A070180 Primes p such that x^3 = 2 has a solution mod p, but x^(3^2) = 2 has no solution mod p.

Original entry on oeis.org

109, 307, 433, 739, 811, 919, 1423, 1459, 1999, 2017, 2143, 2179, 2251, 2287, 2341, 2791, 2917, 2953, 3061, 3259, 3331, 3457, 3889, 4177, 4339, 4519, 4663, 5113, 5167, 5419, 5437, 5653, 6301, 6427, 6661, 6679, 6967, 7723, 7741, 8011, 8389, 8713

Offset: 1

Klaus Brockhaus, Apr 29 2002

Cf. A040028, A049596, A059262, A059667, A070179, A070181 - A070188.

[p: p in PrimesUpTo(10000) | not exists{x: x in ResidueClassRing(p) | x^9 eq 2} and exists{x: x in ResidueClassRing(p) | x^3 eq 2}]; // Vincenzo Librandi, Sep 21 2012

PARI
```
forprime(p=2,8800,x=0; while(x
				
```

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,2,3,3^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

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A049596 Primes p such that x^9 = 2 has a solution mod p.

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A059354 Primes p such that x^27 = 2 has no solution mod p, but x^9 = 2 has a solution mod p.

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A070180 Primes p such that x^3 = 2 has a solution mod p, but x^(3^2) = 2 has no solution mod p.

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