What are the necessary conditions for two of the terms in the Pythagorean triplet $a^2 = b^2 + c^2$ to be prime numbers?
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There is a wellknown parametrization of Phythagorean triples as $k(m^2  n^2)$, $2kmn$ , $k(m^2 + n^2)$ with positive integers $k,m,n$ and $m$ greater $n$. Now, if two are prime we get $k=1$. And also the middle term is never prime. So the question is when are $m^2  n^2$ and $m^2 + n^2$ both prime. The former factors as $(mn) (m+n)$. For this to be prime we need $mn = 1$. So we get two of three are prime if and only if $2n+1$ and $2n^2 + 2n + 1$ are prime for some positive integer $n$. 

