A059264 -id:A059264 - cp's OEIS frontend

A049544 Primes p such that x^12 = 2 has a solution mod p.

2, 23, 31, 47, 71, 89, 113, 127, 167, 191, 223, 233, 239, 257, 263, 281, 311, 353, 359, 383, 431, 439, 479, 503, 593, 599, 601, 617, 647, 719, 727, 743, 839, 863, 881, 887, 911, 919, 983, 1031, 1049, 1097, 1103, 1151, 1193, 1217, 1223, 1289, 1319, 1327

Offset: 1

N. J. A. Sloane

Complement of A059264 relative to A000040. - Vincenzo Librandi, Sep 13 2012

			0^12 == 2 (mod 2). 2^12 == 2 (mod 23). 8^12 == 2 (mod 31). 4^12 == 2 (mod 47). 8^12 == 2 (mod 71). 2^12 == 2 (mod 89). 3^12 == 2 (mod 113). 8^12 == 2 (mod 127). - _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, A059264.

[p: p in PrimesUpTo(1400) | exists(t){x : x in ResidueClassRing(p) | x^12 eq 2}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^12- 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

A059362 Primes p such that x^24 = 2 has no solution mod p.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 79, 83, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 199, 211, 227, 229, 241, 251, 269, 271, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 367

Offset: 1

Klaus Brockhaus, Jan 27 2001

Complement of A049556 relative to A000040.
Coincides for the first 23 terms with sequence A059264 of primes p such that x^12 = 2 has no solution mod p (first divergence is at 113, cf. A059331).
Coincides for the first 161 terms with sequence A212376 of primes p such that x^48 = 2 has no solution mod p (first divergence is at 1217, cf. A059669).

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A000040, A049556, A059264, A059331, A059669, A212376.

[p: p in PrimesUpTo(400) | forall{x: x in ResidueClassRing(p) | x^24 ne 2}]; // Bruno Berselli, Sep 14 2012

Select[Prime[Range[PrimePi[400]]], ! MemberQ[PowerMod[Range[#], 24, #], Mod[2, #]] &] (* Bruno Berselli, Sep 14 2012 *)
ok[p_] := Reduce[Mod[x^24 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[80]], ok] (* Vincenzo Librandi, Sep 20 2012 *)

A059668 Primes p such that x^36 = 2 has no solution mod p, but x^12 = 2 has a solution mod p.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 04 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..357

A000040, A049568, A049544, A059264.

[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

Select[Prime[Range[PrimePi[30000]]], ! MemberQ[PowerMod[Range[#], 36, #], Mod[2, #]]&& MemberQ[PowerMod[Range[#], 12, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 22 2013 *)

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

Original entry on oeis.org

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

Klaus Brockhaus, Apr 29 2002

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

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

[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

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

PARI
```
forprime(p=2,3700,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,2,6,6^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

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

A059331 Primes p such that x^24 = 2 has no solution mod p, but x^12 = 2 has a solution mod p.

Original entry on oeis.org

113, 281, 353, 593, 617, 1049, 1097, 1193, 1481, 1601, 1753, 1889, 2129, 2273, 2281, 2393, 2689, 3089, 3137, 3761, 3833, 4001, 4153, 4217, 4289, 4457, 4657, 4817, 4937, 5113, 5393, 5569, 6521, 6569, 6761, 7481, 7577, 7793, 7817, 7841, 8273, 8369, 8537

Offset: 1

Klaus Brockhaus, Jan 26 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..370

A000040, A049556, A049544, A059264.

[p: p in PrimesUpTo(9000) | not exists{x: x in ResidueClassRing(p) | x^24 eq 2} and exists{x: x in ResidueClassRing(p) | x^12 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Select[Prime[Range[PrimePi[1000]]], !MemberQ[PowerMod[Range[#], 24, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 12, #], Mod[2, #]]&] (* Vincenzo Librandi, Sep 21 2013 *)

A216484 Primes p such that x^36 = 2 has no solution mod p.

Original entry on oeis.org

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

Offset: 1

Vincenzo Librandi, Sep 14 2012

Complement of A049568 relative to A000040.
Different from A059264: 919, 1423, 1999, ... (see A059668) are terms of this sequence, but not of A059264. [Joerg Arndt, Sep 14 2012]
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]

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A049568.

Cf. A059668 (primes p such that x^36 = 2 has no solution mod p, but x^12 = 2 has a solution mod p).

[p: p in PrimesUpTo(400) | forall{x: x in ResidueClassRing(p) | x^36 ne 2}];

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 *)

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A049544 Primes p such that x^12 = 2 has a solution mod p.

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A059362 Primes p such that x^24 = 2 has no solution mod p.

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

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A059331 Primes p such that x^24 = 2 has no solution mod p, but x^12 = 2 has a solution mod p.

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A216484 Primes p such that x^36 = 2 has no solution mod p.

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Mathematica