Re: x^3=3 mod p (cubic reciprocity)

Crios wrote:
Gerry Myerson ha scritto:

(...)

I think OP is asking for criteria to identify those p of the form
6 n + 1 for which there is an integer x such that x^3 = 3 (mod p).

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)

Exactly.
A criterium is well known : 3^((p-1)/6)=+/-1 mod p
but what I mean is a better description of the set in terms of,
as an example, modular constraints, if they exist, or why not.

Let p be a prime of the form 6n+1. Write 4p = A^2 + 27B^2 with A,B
integers and A congruent to 1 mod 3 (that this is always possible was
shown by Gauss--I think it's in the Disquisitiones). Then x^3 == 3
(mod p) has a solution if and only if p^2 + 3p - 4A == 12 (mod 81).

.

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