A Heron Diierence

Is it possible that the lengths of two sides of a primitive Heron triangle have a c ommon factor? The triangle with sides 9, 65, 70 and area 252 shows that this is possible. But notice that the common factor 5 is a prime of the form 4+1. Surprisingly, however, such a common factor cannot be a prime of the form 4 ; 1. Heron's name is familiar to anyone who has used the formula = p s(s ; a)(s ; b)(s ; c) , s = 1 2 (a + b + c) to c alculate the area of a triangle with sides a, b, c. I n a Heron triangle a, b, c, and are positive integers. A H e r on triangle is called primitive when gcd(aabbc) = 1. If a Heron triangle contains a right angle, then it is called a Pythagorean triangle. It is easy t o see that there is no primitive Pythagorean triangle, where two sides have a c ommon factor. The situation for Heron triangles is diierent. The aim of this paper is to p r ove that if there is a common (prime) factor of two sides of a primitive H e r on triangle, then it is always a prime of the form 4 + 1. Deenitions and Observations A Pythagorean triangle is a right triangle with integer sides such as