This is a Diophantine equations problem, which is so interesting one can do by plugging the suitable values in unknown. When it comes for finding set of all solutions is may be tough. I would like to learn to find all solutions of this equations with the help of m.s.e.
If $t \geq 3$ be prime and consider $y^2-x^t= k$ for some fixed integer number $k$. This equation has a solution in $Z_P$ for all $p$ if and only if it has one for primes $p$ of the form $p = 2qt + 1$ with $1 \leqslant q \leqslant (t-3)/2$
Please prove the validity of the above cited statement!