A058853 -id:A058853 - cp's OEIS frontend

A059312 Primes p such that x^59 = 2 has no solution mod p.

709, 827, 1063, 1181, 1889, 2243, 2833, 3187, 3541, 3659, 4013, 4603, 4721, 4957, 5783, 6373, 6491, 7789, 7907, 8969, 9323, 9677, 10267, 10739, 11093, 11329, 11801, 12037, 12391, 13099, 13217, 13807, 14633, 14869, 16993, 19471, 20297, 20533

Offset: 1

Klaus Brockhaus, Jan 25 2001

This is not the same as "Primes congruent to 1 mod 59", A216315. The first missing number is A216315(27) = 11447. - Zak Seidov, Sep 03 2012

Complement of A216886 relative to A000040. - Vincenzo Librandi, Sep 20 2012

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A058853, A216315, A216886.

[p: p in PrimesUpTo(21000) | forall{x: x in ResidueClassRing(p) | x^59 ne 2}]; // Bruno Berselli, Sep 20 2012

Select[Prime[Range[PrimePi[21000]]], ! MemberQ[PowerMod[Range[#], 59, #], Mod[2, #]] &] (* Bruno Berselli, Sep 20 2012 *)
ok[p_] := Reduce[Mod[x^59 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2500]], ok]  (* Vincenzo Librandi, Sep 20 2012 *)

A059230 Primes p such that x^61 = 2 has no solution mod p.

Original entry on oeis.org

367, 733, 977, 1709, 1831, 2441, 3539, 4027, 4271, 4637, 4759, 5003, 5857, 6101, 6833, 7321, 7687, 8053, 8297, 8419, 8663, 9029, 9151, 9883, 10859, 12323, 12689, 13177, 13421, 14153, 14519, 15373, 15739, 16349, 17203, 17569, 18301, 18911

Offset: 1

Klaus Brockhaus, Jan 20 2001

Presumably this is also Primes congruent to 1 mod 61 (A212378). - N. J. A. Sloane, Jul 11 2008
Complement of A216884 relative to A000040. - Vincenzo Librandi, Sep 20 2012
Regarding the first comment, the smallest counterexample is the prime 34039: 34039 == 1 (mod 61), but 1155^61 == 2 (mod 34039), therefore this prime is not in the sequence. - Bruno Berselli, Sep 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A058853, A212378.

[p: p in PrimesUpTo(19000) | forall{x: x in ResidueClassRing(p) | x^61 ne 2} ]; // Vincenzo Librandi, Sep 20 2012

ok[p_]:= Reduce[Mod[x^61 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[5000]], ok] (* Vincenzo Librandi, Sep 20 2012 *)

A059225 Primes p such that x^31 = 2 has no solution mod p.

Original entry on oeis.org

311, 373, 1117, 1303, 1427, 1489, 1861, 2357, 2543, 2729, 2791, 3163, 3659, 3907, 4093, 4217, 4651, 5023, 5147, 5209, 5333, 5519, 5581, 5953, 6263, 6449, 6883, 7069, 7193, 7937, 8123, 8681, 8867, 8929, 9239, 9859, 10789, 11161, 11471, 11657, 11719

Offset: 1

Klaus Brockhaus, Jan 19 2001

Complement of A216883 relative to A000040. - Vincenzo Librandi, Sep 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A058853.

[p: p in PrimesUpTo(12500) | not exists{x : x in ResidueClassRing(p) | x^31 eq 2} ]; // Vincenzo Librandi, Sep 20 2012

 ok[p_]:= Reduce[Mod[x^31 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2000]], ok] (* Vincenzo Librandi, Sep 20 2012  *)

A059243 Primes p such that x^43 = 2 has no solution mod p.

Original entry on oeis.org

173, 431, 947, 1033, 1291, 1549, 1721, 1979, 2237, 2753, 3011, 3527, 3613, 4129, 4817, 4903, 5333, 5591, 5849, 6451, 6709, 6967, 7741, 8171, 8429, 9203, 9461, 9547, 9719, 10321, 10837, 11353, 11783, 12041, 13159, 13331, 13417, 13933, 14621

Offset: 1

Klaus Brockhaus, Jan 21 2001

Complement of A058853 relative to A000040.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A058853.

[p: p in PrimesUpTo(15000) | not exists{x : x in ResidueClassRing(p) | x^43 eq 2} ]; // Vincenzo Librandi, Sep 20 2012

ok[p_]:= Reduce[Mod[x^43 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[4000]], ok] (* Vincenzo Librandi, Sep 20 2012  *)

A059257 Primes p such that x^47 = 2 has no solution mod p.

Original entry on oeis.org

283, 659, 941, 1129, 1223, 1693, 1787, 2069, 2351, 2539, 2633, 3761, 4231, 4513, 4889, 5077, 5171, 5641, 5923, 6299, 6581, 6863, 7333, 8179, 8273, 8461, 8837, 9871, 10247, 10529, 11093, 11657, 11939, 12409, 12503, 12973, 13537, 13913, 14759

Offset: 1

Klaus Brockhaus, Jan 23 2001

Presumably this is also Primes congruent to 1 mod 47. - N. J. A. Sloane, Jul 11 2008.   Not so! The smallest counterexample is 26227: 26227 == 1 (mod 47), but 131^47 == 2 (mod 26227), therefore this prime is not in the sequence. - Bruno Berselli, Sep 12 2012
All terms are 1 mod 94. - Charles R Greathouse IV, Sep 13 2012
Complement of A216885 relative to A000040. - Vincenzo Librandi, Sep 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A058853.

[p: p in PrimesUpTo(15000) | forall{x: x in ResidueClassRing(p) | x^47 ne 2}]; // Bruno Berselli, Sep 12 2012

ok[p_]:= Reduce[Mod[x^47 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[1800]], ok] (* Vincenzo Librandi, Sep 20 2012  *)

select(p->!ispower(Mod(2,p),47),primes(3000)) \\ Charles R Greathouse IV, Sep 13 2012

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A059312 Primes p such that x^59 = 2 has no solution mod p.

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A059230 Primes p such that x^61 = 2 has no solution mod p.

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A059225 Primes p such that x^31 = 2 has no solution mod p.

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A059243 Primes p such that x^43 = 2 has no solution mod p.

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A059257 Primes p such that x^47 = 2 has no solution mod p.

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