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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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If $k$ is odd and greater than $1$, there is none since $a+b$ divides $a^k + b^k$. And every prime of the form $4m+1$ can be expressed as a sum of two squares. – user17762 Sep 27 '12 at 21:28
and if k is even? – fosho Sep 27 '12 at 21:28
If $a=2$ and $b=1$ there are known to be multiple solutions, the so-called Fermat primes. If $a=2$ and $b=3$ then $k=1$, $k=2$ and $k=4$ all give solutions. – Steven Stadnicki Sep 27 '12 at 21:30
see edit please – fosho Sep 27 '12 at 21:31
@fosho Is your question "Given $a$ and $b$, how many primes are of the form $a^k + b^k$?" or "Given $k$, how many primes are of the form $a^k + b^k$?" or is it just "How many primes are of the form $a^k + b^k$"? – user17762 Sep 27 '12 at 21:34
up vote 0 down vote accepted

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.

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