TITLE: Right triangles with integer sides
QUESTION [5 upvotes]: Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as 
$$2x:x^2-1:x^2+1$$
My question is why one of the sides seems to be always prime? (When there is no common divisor)

REPLY [2 votes]: I'll add a comment on why your claim is probably false, even before you find any counterexamples.  Your formula
$$
2x : x^2 - 1 : x^2 + 1
$$
should be a red flag.  $2x$ is never prime for $x > 1$ because it is divisible by 2, and $x^2 - 1 = (x + 1)(x - 1)$ is never prime for $x > 2$ because it factors.  So your claim seems dubious from the start -- only $x^2 + 1$ even has a chance of being prime.