Given that $a^2+b^2=c^2$ and $c$ is prime, is there a value for the shortest side of this right triangle after which the hypotenuse is never again prime? Or is there always a larger prime?
I ask because in the $b+1$ sequence, values for $c=b+1$ begin $5$, $13$, $25$, $41$, $61$, $85$, etc., and primes occur often. But do they occur perpetually? And if they do, is this true for all Pythagorean sequences? Including the $b+2$ sequence? When $c=b+2$ and the value for $a$ is divisible by $4$, the sequence of $c=b+2$ begins $5$, $17$, $37$, $65$, $101$, $145$, $197$, $257$, etc.
Doesn't this mean there is an infinitude of primes of the form $x^2+1$?