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How many primes are there of the form $a^{k/2} + b^{k/2}$ exist for $a$ and $b$ (positive integer solutions).

I am hoping there is only one.

EDIT $k > 1$

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    http://math.stackexchange.com/questions/202247/number-of-solutions-to-equation2012-09-27

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Infinitely many. In fact, every prime $p \equiv 1 \pmod 4$ can be written as the sum of two squares; a result attributed to Fermat. And there are infinitely many such primes, according to Dirichlet's Theorem.