A051071 -id:A051071 - cp's OEIS frontend

A051098 Primes p such that x^58 = -2 has a solution mod p.

2, 3, 11, 17, 19, 41, 43, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683, 691, 739, 761, 769, 787, 809, 811, 827, 857, 859, 881, 883, 907, 937, 947, 953, 971, 977, 1009

Offset: 1

N. J. A. Sloane

Complement of A216774 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(1010) | exists(t){x : x in ResidueClassRing(p) | x^58 eq - 2}]; // Vincenzo Librandi, Sep 16 2012

ok[p_]:= Reduce[Mod[x^58 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[500]], ok] (* Vincenzo Librandi, Sep 16 2012 *)

PARI
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/* see A051071 */
```

More terms from Joerg Arndt, Jul 27 2011

A051099 Primes p such that x^60 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 43, 59, 83, 89, 107, 113, 179, 227, 233, 251, 257, 283, 307, 347, 353, 419, 443, 467, 499, 563, 587, 593, 617, 643, 659, 683, 739, 827, 947, 971, 1019, 1049, 1097, 1163, 1187, 1193, 1217, 1259, 1283, 1289, 1307, 1427, 1433, 1459, 1499, 1523, 1553, 1579

Offset: 1

N. J. A. Sloane

Complement of A216775 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

Cf. A000040, A216775.

[p: p in PrimesUpTo(1600) | exists(t){x : x in ResidueClassRing(p) | x^60 eq - 2}]; // Vincenzo Librandi, Sep 16 2012

ok[p_]:= Reduce[Mod[x^60 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[500]], ok] (* Vincenzo Librandi, Sep 16 2012 *)

PARI
```
/* see A051071 */
```

More terms from Joerg Arndt, Jul 27 2011

A216690 Primes p such that x^4 = -2 has no solution mod p.

Original entry on oeis.org

5, 7, 13, 17, 23, 29, 31, 37, 41, 47, 53, 61, 71, 79, 97, 101, 103, 109, 127, 137, 149, 151, 157, 167, 173, 181, 191, 193, 197, 199, 223, 229, 239, 241, 263, 269, 271, 277, 293, 311, 313, 317, 349, 359, 367, 373, 383, 389, 397

Offset: 1

Vincenzo Librandi, Sep 15 2012

Complement of A051071 relative to A000040.

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

Cf. A000040, A051071.

ok[p_]:= Reduce[Mod[x^4 + 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]], ok]

A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

Original entry on oeis.org

281, 617, 1033, 1049, 1097, 1193, 1481, 1553, 1753, 1777, 2281, 2393, 2473, 2657, 2833, 2857, 3049, 3529, 3673, 3833, 4049, 4153, 4217, 4273, 4457, 4937, 5113, 5297, 5881, 6121, 6449, 6481, 6521, 6529, 6569, 6761, 6793, 6841, 7121, 7129, 7481, 7577, 7817, 8081, 8233, 8537, 9001, 9137, 9209, 9241

Offset: 1

Christopher J. Smyth, Jul 22 2009

Each term p has the form 2^r*j + 1, where r >= 3, j is odd, and ord_p(-2) divides j.

			281 is in the sequence as 281 = 2^3*35 + 1 and 281 | 2^35 + 1.

Set difference of A163183 and A007520.

Subsequence of A033203, A051071, A051073, A051077, A051085, A051101.

with(numtheory):A:=NULL:p:=2: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 and p mod 8 = 1 then A:=A,p;;fi;od:A;

More terms from Max Alekseyev, Sep 29 2016

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A051098 Primes p such that x^58 = -2 has a solution mod p.

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A051099 Primes p such that x^60 = -2 has a solution mod p.

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A216690 Primes p such that x^4 = -2 has no solution mod p.

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A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

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