Counting Heron triangles with constraints

Abstract

Heron triangles have the property that all three of their sides as well as their area are positive integers. In this paper, we give some estimates for the number of Heron triangles with two of their sides fixed. We provide a general bound on this count H (a,b), where the sides a,b are fixed positive integers, and the estimate here is better than the one of Ionescu, Luca and Stanica for the general situation of fixed sides a,b. In the case of primes sides p,q, there is an additional hypothesis which helps us to drop the upper bounds on H (p.q). In particular, we prove that H(p,q) is less than or equal to 1 when p - q = 2 (mod 4). We also provide a count for the number of Heron triangles with a fixed height (there exists only one such when the height is prime. Moreover, we study the decomposability property of a Heron triangle into two similar ones, and provide some cases when a Heron triangle is not decomposable.