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., it 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.
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