Classification and Enumeration of Primitive Eisenstein Triples using Prime Factorization Techniques

Abstract

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  $\gcd(a, b, c) = 1$. 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.