I've been discussing a problem recently
Let $p, q$ be primes. If $x^q\equiv2\pmod p$ has no solution then $p\equiv1\pmod q.$
This is not a bi-equivalence (though it is "nearly" one): there are 811 primes below a million which are 1 mod $q=97,$ of which $x^q\equiv2\pmod p$ has no solution in 803 cases. (There are no cases below a million without a solution unless $p\equiv1\pmod{97}.$) There's nothing special about 97; other primes have similar behavior.
So:
- Is the statement true?
- Can the exceptional primes ($p\equiv1\pmod q$ but $x^q\equiv2\pmod p$ has a solution) be characterized?