Concerning a Class of Diophantine Equations with Ordinal Variables

It has been shown by Sierpiński that the Diophantine x 3 = y 3 + 1 has no solution in transfinite ordinals. A short time later, Świerczkowski generalized this result by demonstrating the non‐existance of transfinite solutions of the equation x α = y rβ + k , where k is a positive integer and α,rβ are successon ordinals = 1. We commence this note with a simple demonstration that the restriction of α,rβ to successor ordinals was unnecessary, and then go on to consider the existence or non‐existence of transfinite solution of a related class of Diophantine equations.