[seqfan] Re: Two questions looking for answers

[Don Reble]

> A045318 Primes p such that x^8 = 3 has no solution mod p.
> A301916 Primes which divide numbers of the form 3^k + 1.

> The terms in A301916 but not in A045318 are ... A320481.
>     2, 769, 1297, 6529, 7057, 8017, 8737, 12097, 12289. ...
> ...what are these primes?

    Let POT(x) be the largest power of 2 which divides x.
    The non-2 values are those primes P, such that
	2 <= POT(the order of 3 modulo P) <= POT(P-1)/8.
    "2 <=" so that P divides some 3^k+1,
    "<= POT(P-1)/8" so that 3 has an eighth root modulo P.

    POT = A006519; A062117 is the order of 3 modulo prime(n).

-- 
Don Reble  djr at nk.ca