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S.E board! 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 number is $k$ from set of integers. 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!

Advanced thanks.

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