Pythagorean triangles with legs less than n

Abstract

We obtain asymptotic estimates for the number of Pythagorean triples ( a, b, c) such that a<n, b<n . These estimates (considering the triple ( a, b, c) different from ( b, a, c)) is (4π −2 log(1+ 2 ))n+ O( n ) in the case of primitive triples, and (4π −2 log(1+ 2 ))n log n+ O(n) in the case of general triples. Furthermore, we derive, by a self-contained elementary argument, a version of the first formula which is weaker only by a log-factor. Also, we tabulate the number of primitive Pythagorean triples with both legs less than n, for selected values of n⩽1 000 000 000 , showing the excellent precision obtained.