Characterization of Diophantine Equations <i>a</i> + <i>y</i> ² = <i>z</i> ² , Pythagorean <i>n</i> ‐Tuples, and Algebraic Structures

Let , and 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 + y 2 = z 2 . 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 + y 2 = z 2 with elements in . Additionally, we intend to establish a commutative infinite monoid with elements in or on the set of Pythagorean quadruples. Moreover, in the set of Pythagorean quadruples, we aim to find a commutative infinite group with elements in or . To achieve these results, we prove the existence of some suitable binary operations.