OFFSET
1,1
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.
LINKS
Jinyuan Wang, Table of n, a(n) for n = 1..1000
EXAMPLE
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).
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;
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Christopher J. Smyth, Jul 24 2009
EXTENSIONS
More terms from Jinyuan Wang, Mar 24 2020
STATUS
approved