This is my guess , i hope that's true .

Let's say a triangle is a Heron triangle if the lengths of its three sides as well as its area

are integers.

I guess : If $\displaystyle p ,q $ are primes such that $\displaystyle p^2 + 1 = 2q $ , then all the non-isosceles Heron triangles with $\displaystyle p$ as the length of one of the sides are right-angled . Therefore , we can only find out one or two possible Heron triangle(s) for it .

For example , let $\displaystyle p = 5 $ we have $\displaystyle 5^2 + 1 = 2(13) $

Then the only Heron triangles are $\displaystyle (3,4,5) ~ ,~ (5,12,13) $ ?