In this paper, we study the function H(a,b), which associates to every pair of positive integers a and b and the number of positive integers c such that the triangle sides of 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 adn b.
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