The generalized Pellian equation

Abstract

It is known from number theory that the rational integral solutions of the Diophantic equation in two unknowns of second degree, viz.

Unit Fractions and the Erdã–s-Straus Conjecture

“…When x,y,z are distinct the solution represents an Egyptian fraction for 4/n [25]. Sometimes there are multiple solutions such as when n = 5, (x,y,z) = (2,4,20) and (2,5,10). This case makes us wonder if there are patterns, and there are for some n, but not for all n [14].…”

“…Continued fractions can be used in integer structures for rational approximations of real numbers and Diophantine equations with second order linear recurrence relations [4,19,20] as can their multidimensional generalizations with arbitrary order linear recurrence relations [22]. These computational exercises lead quite naturally into topics in the philosophy of mathematics [5] as foreshadowed in the Introduction to this note.…”