Heron triangles with two fixed sides

In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is Heron, i.e., has integral area. In particular, we prove that $H(p,q)\le 5$ if $p$ and $q$ are primes, and that $H(a,b)=0$ for a random choice of positive integers $a$ and $b$.Comment: 20 pges, no figures, submitted in 200

Heron triangles with two fixed sides

In this paper, we study the function H(a, b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., has integral area. In particular, we prove that H(p, q) ≤ 5 if p and q are primes, and that H(a, b) = 0 for a random choice of positive integers a and b