Pythagorean triangles with legs less than n

摘要

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~(1/2))n + O(n~(1/2)) in the case of primitive triples, and (4π~(-2) log(1 + 2~(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.