I put together a VBA script to find me all of the Pythagorean Triangles from 3,4,5 all the way to 105, 608, 617. Then I arranged them by increasing hypotenuse $c$, and identified all of the triangles where $c$ was a prime number. I then subtracted each $c$ value from the preceeding prime $c$ and found that in every instance the difference between the two values was divisible by 4.
and $41-37 \equiv 0 \pmod 4$.
and $53-41 \equiv 0 \pmod 4$. And so forth.
Is there a way to prove whether or not this holds true for all prime numbers that can be represented by the square root of the sum of two square integers?