A Remark on Prim Divisors of Lengths of Sides of Heron Triangles

Abstract

A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. Let {\small ${\cal P}$} be a fixed set of primes; let S denote the set of integers divisible only by primes in {\small ${\cal P}$}. We prove that there are only finitely many Heron triangles whose sides {\small $a,b,c\in S$} and are reduced, that is {\small $\gcd (a,b,c)=1$}. If {\small ${\cal P}$} contains only one prime {\small $\equiv 1\; (\bmod \; 4)$}, then all these triangles can be effectively determined. In case {\small ${\cal P}=\{2,3,5,7,11\}$}, all such triangles are explicitly given.