(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This study delves into the concept of primitive Eisenstein triples, defined as positive integer solutions (a, b, c) to the quadratic equation a^2 - ab + b^2 = c^2, subject to the condition a<c<b and (a, b, c) = 1. We classify these triples according to the prime factorization of the integer c, elucidating how their existence is intricately linked to specific congruence conditions imposed on the prime divisors of c. Furthermore, we establish a bijective correspondence between these triples and a certain subset of the unit circle. This correspondence enables a comprehensive enumeration of the triples and precisely characterizes the conditions under which such solutions exist.