A059362 -id:A059362 - cp's OEIS frontend

A059264 Primes p such that x^12 = 2 has no solution mod p.

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

Klaus Brockhaus, Jan 23 2001

Complement of A049544 relative to A000040.

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).

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

Cf. A000040, A049544, A049568, A059668.

ok[p_] := Reduce[Mod[x^12 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]],ok] (* Vincenzo Librandi, Sep 14 2012 *)
Select[ Prime@ Range@ PrimePi@400, !MemberQ[ PowerMod[ Range@#, 12, #], Mod[2, #]] &] (* Robert G. Wilson v, Nov 05 2016 after Bruno Berselli in A059362 *)

A049556 Primes p such that x^24 = 2 has a solution mod p.

Original entry on oeis.org

2, 23, 31, 47, 71, 89, 127, 167, 191, 223, 233, 239, 257, 263, 311, 359, 383, 431, 439, 479, 503, 599, 601, 647, 719, 727, 743, 839, 863, 881, 887, 911, 919, 983, 1031, 1103, 1151, 1217, 1223, 1289, 1319, 1327, 1367, 1399, 1423, 1433, 1439, 1471, 1487, 1511

Offset: 1

N. J. A. Sloane

Complement of A059362 relative to A000040. - Vincenzo Librandi, Sep 14 2012

			0^24 == 2 (mod 2). 5^24 == 2 (mod 23). 3^24 == 2 (mod 31). 2^24 == 2 (mod 47). 24^24 == 2 (mod 71). 25^24 == 2 (mod 89). 5^24 == 2 (mod 127). 63^24 == 2 (mod 167). - _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, A059362.

[p: p in PrimesUpTo(1600) | exists(t){x : x in ResidueClassRing(p) | x^24 eq 2}]; // Vincenzo Librandi, Sep 14 2012

ok[p_]:= Reduce[Mod[x^24 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[300]], ok] (* Vincenzo Librandi, Sep 14 2012 *)

A212376 Primes p such that x^48 = 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

Bruno Berselli, Sep 14 2012

Complement of A049580 relative to A000040.

This sequence is not the same as A059362. First disagreement at index 162: a(162)=1217, A059362(162)=1229.

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

Cf. A000040, A049580.

Cf. A059362.

[p: p in PrimesUpTo(400) | forall{x: x in ResidueClassRing(p) | x^48 ne 2}];

Select[Prime[Range[PrimePi[400]]], ! MemberQ[PowerMod[Range[#], 48, #], Mod[2, #]] &]
ok[p_] := Reduce[Mod[x^48 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[75]], ok] (* Vincenzo Librandi, Sep 21 2012 *)

N=10^4;  default(primelimit,N);
ok(p, r, k)={ return ( (p==r) || (Mod(r,p)^((p-1)/gcd(k,p-1))==1) ); }
forprime(p=2,N, if (! ok(p,2,48),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

A059669 Primes p such that x^48 = 2 has no solution mod p, but x^24 = 2 has a solution mod p.

Original entry on oeis.org

1217, 1553, 1777, 2833, 4049, 4481, 5297, 6449, 6689, 7121, 8081, 8609, 9137, 9281, 9649, 10337, 10433, 11329, 11633, 12241, 13121, 14321, 14753, 15569, 16433, 16673, 18257, 19793, 23057, 25169, 25889, 26177, 26561, 26993, 27281, 28001, 29153, 29201

Offset: 1

Klaus Brockhaus, Feb 04 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..188

Cf. A000040, A049556, A059362, A049580.

[p: p in PrimesUpTo(30000) | not exists{x: x in ResidueClassRing(p) | x^48 eq 2} and exists{x: x in ResidueClassRing(p) | x^24 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Select[Prime[Range[PrimePi[30000]]], ! MemberQ[PowerMod[Range[#], 48, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 24, #], Mod[2, #]] &] (* Vincenzo Librandi, Sep 22 2013 *)

a(37)-a(38) from Vincenzo Librandi, Sep 21 2012

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A059264 Primes p such that x^12 = 2 has no solution mod p.

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A049556 Primes p such that x^24 = 2 has a solution mod p.

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A212376 Primes p such that x^48 = 2 has no solution mod p.

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A059669 Primes p such that x^48 = 2 has no solution mod p, but x^24 = 2 has a solution mod p.

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