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Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures

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  • Roberto Amato

Abstract

Let N,Z, and Q be the sets of natural, integers, and rational numbers, respectively. Our objective, involving a predetermined positive integer a, is to study a characterization of Diophantine equations of the form a+y2=z2. Building on this result, we aim to obtain a characterization for Pythagorean n-tuples. Furthermore, we seek to prove the existence of a commutative infinite monoid in the set of Diophantine equations a+y2=z2 with elements in N. Additionally, we intend to establish a commutative infinite monoid with elements in N or Z on the set of Pythagorean quadruples. Moreover, in the set of Pythagorean quadruples, we aim to find a commutative infinite group with elements in Q or Z. To achieve these results, we prove the existence of some suitable binary operations.

Suggested Citation

  • Roberto Amato, 2025. "Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2025, pages 1-19, May.
    • Handle: RePEc:hin:jijmms:5516311
    DOI: 10.1155/ijmm/5516311
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