On the algebraic structure of Pythagorean triples

Autori

DOI:

https://doi.org/10.1478/AAPP.1021A3

Parole chiave:

Pythagorean triples. Groups. Irrationality

Abstract

A Pythagorean triple is an ordered triple of integers (a,b,c) ≠ (0, 0, 0) such that a2 + b2 = c2. It is well known that the set ℘ of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation (℘,⋆). In this article, we will introduce the "commensurability" relation ℛ among Pythagorean triples, and we will see that it induces a group quotient, ℘/ℛ, which is isomorphic with the direct product of infinite (countable) copies of C∞, the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees.

Biografie autore

  • Giuseppina Anatriello, Università degli Studi di Napoli "Federico II"
    Dipartimento di Architettura
  • Giovanni Vincenzi, Università degli Studi di Salerno
    Dipartimento di Matematica

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Pubblicato

2024-02-10

Fascicolo

V. 102 N. 1 (2024)

Sezione

Articoli

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