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

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

Author

Christopher J. Smyth, Jul 24 2009

Keywords

• nonn
• easy

Comments

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.

Examples

			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).
		

Links

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

Crossrefs

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.

Programs

Maple

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;

PARI

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

Extensions

More terms from Jinyuan Wang, Mar 24 2020