A163183 -id:A163183 - cp's OEIS frontend

A051101 Primes p such that x^64 = -2 has a solution mod p.

2, 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, 1123, 1163, 1171, 1187

Offset: 1

N. J. A. Sloane

Differs from A051085 first at the 541st entry, at p=15809. - R. J. Mathar, Oct 14 2008
From Christopher J. Smyth, Jul 24 2009: (Start)
Differs from A163183 (primes dividing 2^j+1 for some odd j) at the 827th entry, at p=25601. See comment at A163186 for explanation.
Sequence is union of A163183 and A163186 (primes p such that the equation x^64 = -2 mod p has a solution, and ord_p(-2) is even).
(End)
Complement of A216777 relative to A000040. - Vincenzo Librandi, Sep 17 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

[p: p in PrimesUpTo(1200) | exists(t){x : x in ResidueClassRing(p) | x^64 eq - 2}]; // Vincenzo Librandi, Sep 16 2012

ok[p_]:= Reduce[Mod[x^64 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[400]], ok] (* Vincenzo Librandi, Sep 16 2012 *)

forprime(p=2, 2000, if([]~!=polrootsmod(x^64+2, p), print1(p, ", "))); print();
/* Joerg Arndt, Jun 24 2012 */

More terms from Joerg Arndt, Jul 27 2011

A163185 Primes p such that the equation x^2 = -2 mod p has a solution, and ord_p(-2) is even.

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1129, 1153, 1201, 1217, 1249, 1289, 1297, 1321, 1361, 1409, 1433, 1489, 1601, 1609

Offset: 1

Christopher J. Smyth, Jul 23 2009

Such primes are the exceptional p for which x^2 == -2 (mod p) has a solution, as x^2 == -2 (mod p) is soluble for *every* p with ord_p(-2) odd. But if ord_p(-2) is even and p - 1 = 2^r.j with j odd, then x^2 == -2 (mod p) is soluble if and only if ord_p(-2) is not divisible by 2^r.
More generally, the equation x^(2^k) == -2 (mod p) has a solution iff either ord_p(-2) is odd or (p == 1 (mod 2^(k+1)) and ord_p(-2) is even but not divisible by 2^(r-k+1)).
Proof: Choose primitive root g mod p with -2 == g^a (mod p), where a = (p-1)/ord_p(-2). Writing x = g^u, see that solving x^(2^k) == -2 (mod p) is equivalent to solving u*2^k + v*(p-1) = a for some integers u,v.
A necessary and sufficient condition for this is that gcd(2^k,p-1) | a. So for p-1 = j*2^r, j odd and ord_p(-2) = h*2^s, h odd, condition becomes min(k,r) <= r-s. If s = 0 (i.e., ord_p(-2) odd) this is always valid; for positive s we need k < r-s+1, or s < r-k+1.

			17 belongs to this sequence as 7^2 == -2 (mod 17) and ord_p(-2) = 8, even but <> 0 (mod 16).

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A033203 (all p for which x^2 == -2 (mod p) has a solution); .

Cf. A163183 (p with ord_p(-2) odd): a subsequence of A033203, whose complement in A163183 is the current sequence.

with(numtheory):k:=1: A:=NULL:p:=2: for c to 30000 do p:=nextprime(p);o:=order(-2,p);R:=gcd(2^100,p-1);if o mod 2=0 and p mod 2^(k+1) = 1 and o mod R/2^(k-1)<>0 then A:=A,p;;fi;od:A;

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2==0 && []~!=polrootsmod(x^2+2, p), print1(p, ", "))); \\ Jinyuan Wang, Mar 24 2020

More terms from Jinyuan Wang, Mar 24 2020

A163186 Primes p such that the equation x^64 == -2 (mod p) has a solution, and ord_p(-2) is even.

Original entry on oeis.org

25601, 50177, 59393, 65537, 96001, 115201, 195457, 262657, 266369, 267649, 279553, 286721, 295937, 299393, 306689, 331777, 366593, 425857, 460289, 495617, 509569, 525313, 528001, 556289, 595073, 710273, 730753, 790529, 819457, 868993, 1021697, 1022977, 1049089

Offset: 1

Christopher J. Smyth, Jul 24 2009

Such primes are the exceptional p for which x^64 == -2 (mod p) has a solution, as x^64 == -2 (mod p) is soluble for *every* p with ord_p(-2) odd.
But if ord_p(-2) is even and p - 1 = 2^r.j with j odd, then x^64 == -2 (mod p) is soluble if and only if ord_p(-2) is not divisible by 2^(r-5). See comment at A163185 for explanation.
Most primes p for which x^64 == -2 (mod p) has a solution (A051101) have ord_p(-2) odd (so belong to A163183). Thus 25601 (first element of current sequence, and 827th element of A051101) is the first element where A051101 and A163183 differ.

			For p = 25601, 562^64 == -2 (mod p), the 2-power part of p-1 is 2^10 and ord_p(-2) = 400, which is even but has 2-power part 2^4, which is not divisible by 2^(10-5).

Jinyuan Wang, Table of n, a(n) for n = 1..1000

A051101 (all primes p for which x^62 == -2 (mod p) has a solution) is a union of A163183 (primes p with ord_p(-2) odd) and the current sequence.

with(numtheory):k:=6: A:=NULL:p:=2: for c to 30000 do p:=nextprime(p); o:=order(-2, p); R:=gcd(2^100, p-1); if o mod 2=0 and p mod 2^(k+1) = 1 and o mod R/2^(k-1)<>0 then A:=A, p; fi; od:A;

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2==0 && []~!=polrootsmod(x^64+2, p), print1(p, ", "))); \\ Jinyuan Wang, Mar 24 2020

More terms from Jinyuan Wang, Mar 24 2020

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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A051101 Primes p such that x^64 = -2 has a solution mod p.

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A163185 Primes p such that the equation x^2 = -2 mod p has a solution, and ord_p(-2) is even.

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A163186 Primes p such that the equation x^64 == -2 (mod p) has a solution, and ord_p(-2) is even.

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

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Examples

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Maple

Extensions