Counting Heron triangles with constraints (Q2855575)

A Heron triangle is a triangle whose sides are area are all integers. In the paper under review, the authors fix two positive integers \(a\) and \(b\), and give upper bounds for the function \(H(a,b)\) which counts the number of Heron triangles with two fixed sides \(a\) and \(b\). They obtain that if the factorization of \(ab\) is \(ab=2^{\alpha_0} \prod_{i=1}^{s} p_i^{a_i} \prod_{j=1}^r q_j^{e_j}\), where \(p_i\equiv 3\pmod 4\) and \(q_j\equiv 1\pmod 4\) are primes, then NEWLINE\[CARRIAGE_RETURNNEWLINE H(a,b)\leq \frac{3+(-1)^{ab}}{2} \left(\prod_{j=1}^r (2e_j+1) -1\right). CARRIAGE_RETURNNEWLINE\]NEWLINE In particular, \(H(ab)=0\) if all prime factors of \(ab\) are congruent to \(3\) modulo \(4\), and \(H(p,q)\leq 8\) if \(p\) and \(q\) are primes. They prove a better estimate for \(H(p,q)\) according to the residue classes of \(p\) and \(q\) modulo \(4\). They also show that \(H(p,q)=0\) for special pairs of primes, such as when \(p=2q+1\) (Sophie Germain primes) or when \(p\) and \(q\) are both Mersenne primes. They also prove some other results concerning Heron triangles such as that there are no isosceles Heron triangles with square sides and give a lower bound for the number of Heron triangles with a fixed integer height \(h\).