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

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.