A forest of Eisensteinian triplets

Abstract

In 1934 B. Berggren first discovered the surprising result that every Pythagorean triplet is the pre product of the triplet (3, 4, 5) presented as a column by a product of three matrices, that every triplet is obtained in this manner exactly once and in primitive form. In this paper we show a similar result for integer triangles with an angle of 60 degrees (also known as Eisensteinian triplets). We show that any such triangle is obtained by pre-multiplication (7,8,5) or (13,15,7) by a product of five matrices. The result might have applications in enumerating points with rational distance from the origin on the hexagonal lattice.