On the algebraic structure of Pythagorean triples
A Pythagorean triple is an ordered triple of integers (a,b,c) ≠ (0, 0, 0) such that a^2 + b^2 = c^2. 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 &...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Accademia Peloritana dei Pericolanti
2024-02-01
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| Series: | Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali |
| Online Access: |
http://dx.doi.org/10.1478/AAPP.1021A3
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| Summary: | A Pythagorean triple is an ordered triple of integers (a,b,c) ≠ (0, 0, 0) such that a^2 + b^2 = c^2. 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. |
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| ISSN: | 0365-0359 1825-1242 |