Pythagorean triangles with legs less than n

  1. Benito, M. 1
  2. Varona, J.L. 2
  1. 1 Departamento de Matemáticas, Instituto Práxedes Mateo Sagasta, Dr. Zubía s/n, 26003 Logroño, Spain
  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Journal:
Journal of Computational and Applied Mathematics

ISSN: 0377-0427

Year of publication: 2002

Volume: 143

Issue: 1

Pages: 117-126

Type: Article

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DOI: 10.1016/S0377-0427(01)00496-4 SCOPUS: 2-s2.0-0036608367 WoS: WOS:000176146300009 GOOGLE SCHOLAR lock_openOpen access editor

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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. © 2002 Elsevier Science B.V. All rights reserved.