Question concerning non-similar right triangles with integer lengths

Question

Let $A$ be the set of all $d$ such that $d<1$ and $d=\frac ab$ where $a^2+b^2=c^2$ and $a,b,c \in \Bbb I$. Is $A$ an infinite or finite set and how can that be proven?

Answer 1

Well, Pythagorean triples are given by

$$(a,b,c)=(2pq,p^2-q^2,p^2+q^2)$$

for integers $p$ and $q$ with $p>q$ (to prove that these are all is irrelevant; we only need infinitely many triples, not necessarily all of them).

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So let $q=1$ and $p$ an odd prime, then we have the triple $(2p,p^2-1,p^2+1)$. The corresponding $d$ would then be $d=\frac{2p}{p^2-1}$ (since $2p<p^2-1$ for $p>2$), this is an irreducible fraction if we divide both sides by $2$ (that is $\frac{p}{(p^2-1)/2}$, and this is irreducible because $p^2-1$ and $p$ are coprime) and so it's different for every $p$. Since there are infinitely many primes $p$, the set $A$ is infinitely large (it actually contains a lot more numbers than we've shown, but that is irrelevant for it infinite-ness).