A051071 -id:A051071 - cp's OEIS frontend

A163183 Primes dividing 2^j + 1 for some odd j.

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 281, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 617, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1033, 1049, 1051, 1091, 1097

Offset: 1

Christopher J. Smyth, Jul 22 2009

Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).
All such p are = 1 or 3 mod 8, so sequence is subsequence of A033200, as (-2)^{j+1} == -2 (mod p) implies that (-2/p) = 1, p == 1 or 3 (mod 8).
Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.
Proof: If p = 8r + 3 then 2^{4r+1} == 1 or -1 (mod p). If former, then (2^{2r+1})^2 == 2 (mod p), (2/p) = 1, only true for p == 1 or 7 (mod 8). So p | 2^{4r+1} + 1.
Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.
Claim: For every p in sequence and every 2^k, the equation x^{2^k} == -2 (mod p) is soluble. Hence sequence is a subsequence of A033203 (k=1), A051071 (k=2), A051073 (k=3), A051077 (k=4), A051085 (k=5), A051101 (k=6), ....
Proof: Put x == (-2)^u (mod p). Then using (-2)^j == 1 (mod p), we can solve x^{2^k} == -2 (mod p) if can find u and v such that u*2^k + v*j = 1, possible as gcd(2^k, j) = 1.
From Jianing Song, Jun 22 2025: (Start)
The multiplicative order of -2 modulo a(n) is A385228(n).
Contained in primes congruent to 1 or 3 modulo 8 (primes p such that -2 is a quadratic residue modulo p, A033200), and contains primes congruent to 3 modulo 8 (A007520).
Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)

			11 is in sequence as 11 | 2^5 + 1; 281 (smallest element of the sequence == 1 (mod 8)) is in the sequence as 281 | 2^35 + 1.

Jianing Song, Table of n, a(n) for n = 1..10000

Sequence is a union of A007520 and A163184.
Subsequence of A033200. Contains A007520 as a subsequence.
Cf. A385228 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), this sequence (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

with(numtheory):A:=3:p:=3: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 then A:=A,p;;fi;od:A;

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* Paolo Xausa, Jun 30 2025 *)

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020

A051100 Primes p such that x^62 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 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, 929, 937, 947, 953, 971, 977, 1009

Offset: 1

N. J. A. Sloane

Differs from A033203 first at the 109th entry, at p=1427. - R. J. Mathar, Oct 14 2008

Complement of A216776 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^62 eq - 2}]; // Vincenzo Librandi, Sep 16 2012

ok[p_]:= Reduce[Mod[x^62 + 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

A051076 Primes p such that x^14 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 19, 41, 59, 67, 73, 83, 89, 97, 107, 131, 137, 139, 163, 179, 193, 227, 233, 241, 251, 257, 283, 307, 313, 331, 347, 353, 401, 409, 419, 433, 443, 457, 467, 499, 521, 523, 563, 569, 571, 577, 587, 593, 601, 619, 641, 643, 673, 683, 691, 739, 761, 769, 787, 809, 811, 857, 859, 881, 907, 929, 937, 947, 953, 971, 977, 1019

Offset: 1

N. J. A. Sloane

Complement of A216738 relative to A000040. - Vincenzo Librandi, Sep 16 2012

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

[p: p in PrimesUpTo(1030) | exists(t){x : x in ResidueClassRing(p) | x^14 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^14 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[400]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051078 Primes p such that x^18 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 41, 43, 59, 83, 89, 107, 113, 131, 137, 179, 227, 233, 251, 257, 281, 283, 347, 353, 401, 419, 443, 449, 457, 467, 491, 499, 521, 563, 569, 587, 593, 601, 617, 641, 643, 659, 683, 691, 761, 809, 827, 857, 881, 929, 947, 953, 971, 977, 1019, 1049, 1051, 1091, 1097, 1163, 1187, 1193, 1217, 1259, 1283, 1289, 1307, 1361, 1409

Offset: 1

N. J. A. Sloane

Differs from A051096 first at the 640th entry, at p=17659, next at p=23059. - R. J. Mathar, Oct 14 2008

Complement of A216740 relative to A000040. - Vincenzo Librandi, Sep 16 2012

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

[p: p in PrimesUpTo(1410) | exists(t){x : x in ResidueClassRing(p) | x^18 eq - 2}]; // Vincenzo Librandi Sep 15 2012

ok[p_]:= Reduce[Mod[x^18 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[400]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051079 Primes p such that x^20 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 19, 43, 59, 67, 73, 83, 89, 107, 113, 139, 163, 179, 227, 233, 251, 257, 283, 307, 337, 347, 353, 379, 419, 443, 467, 499, 523, 547, 563, 571, 577, 587, 593, 617, 619, 643, 659, 683, 739, 787, 827, 859, 883, 907, 937, 947, 971, 1019, 1033, 1049, 1097, 1123

Offset: 1

N. J. A. Sloane

Complement of A216741 relative to A000040. - Vincenzo Librandi, Sep 16 2012

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

[p: p in PrimesUpTo(1150) | exists(t){x : x in ResidueClassRing(p) | x^20 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^20 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[400]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051080 Primes p such that x^22 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 19, 41, 43, 59, 73, 83, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 379, 401, 409, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 619, 641, 643, 659, 673, 691, 739, 761, 769, 787, 809, 811, 827, 857, 883, 907, 929, 937, 953, 971, 977, 1009

Offset: 1

N. J. A. Sloane

Complement of A216742 relative to A000040. - Vincenzo Librandi, Sep 16 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^22 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^22 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[300]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051081 Primes p such that x^24 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 43, 59, 83, 107, 131, 179, 227, 251, 257, 281, 283, 307, 347, 419, 443, 467, 491, 499, 563, 587, 617, 643, 659, 683, 691, 739, 811, 827, 881, 947, 971, 1019, 1049, 1051, 1091, 1097, 1163, 1187, 1193, 1217, 1259, 1283, 1307, 1427, 1451, 1459, 1481

Offset: 1

N. J. A. Sloane

Complement of A216743 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(1500) | exists(t){x : x in ResidueClassRing(p) | x^24 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

isA051081 := proc(p) local x; for x from 0 to p-1 do if (x^24 mod p) = (-2 mod p) then RETURN(true) ; fi; od: RETURN(false) ; end: for i from 1 to 300 do p := ithprime(i) ; if isA051081(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Oct 15 2008

ok[p_]:= Reduce[Mod[x^24 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[500]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from R. J. Mathar, Oct 15 2008

A051082 Primes p such that x^26 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 331, 337, 347, 353, 379, 401, 409, 419, 433, 449, 457, 467, 491, 499, 523, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683, 691, 739, 761, 769, 787, 809, 811, 827, 857, 881, 883, 907, 929, 947, 953, 971, 977, 1009

Offset: 1

N. J. A. Sloane

Complement of A216744 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^26 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^26 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[300]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051083 Primes p such that x^28 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 19, 59, 67, 73, 83, 89, 107, 131, 139, 163, 179, 227, 233, 251, 257, 283, 307, 331, 347, 353, 419, 443, 467, 499, 523, 563, 571, 577, 587, 593, 601, 619, 643, 683, 691, 739, 787, 811, 859, 881, 907, 937, 947, 971, 1019, 1033, 1049, 1091, 1097, 1123, 1153, 1163, 1171, 1187

Offset: 1

N. J. A. Sloane

Complement of A216745 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(1200) | exists(t){x : x in ResidueClassRing(p) | x^28 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^28 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[500]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

A051084 Primes p such that x^30 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 17, 43, 59, 83, 89, 107, 113, 137, 179, 227, 233, 251, 257, 283, 307, 347, 353, 419, 433, 443, 449, 457, 467, 499, 563, 569, 587, 593, 617, 641, 643, 659, 683, 739, 809, 827, 857, 929, 947, 953, 971, 977, 1019, 1049, 1097, 1163, 1187, 1193, 1217, 1259, 1283, 1289, 1307, 1409

Offset: 1

N. J. A. Sloane

Complement of A216746 relative to A000040. - Vincenzo Librandi, Sep 17 2012

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

[p: p in PrimesUpTo(1410) | exists(t){x : x in ResidueClassRing(p) | x^30 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^30 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[500]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

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

More terms from Joerg Arndt, Jul 27 2011

cp's OEIS Frontend

A163183 Primes dividing 2^j + 1 for some odd j.

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A051100 Primes p such that x^62 = -2 has a solution mod p.

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A051076 Primes p such that x^14 = -2 has a solution mod p.

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A051078 Primes p such that x^18 = -2 has a solution mod p.

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A051079 Primes p such that x^20 = -2 has a solution mod p.

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A051080 Primes p such that x^22 = -2 has a solution mod p.

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A051081 Primes p such that x^24 = -2 has a solution mod p.

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