The Equations $2^n \pm 2^m \pm 1 = x^2$ in the Arithmetic of the Even and Odd

TL;DR

This paper demonstrates that solutions to specific exponential equations involving powers of two and squares can be derived within a weak arithmetic framework, extending elementary methods with hypergeometric function results.

Contribution

It shows that solutions to the equations $2^n \u00b1 2^m \u00b1 1 = x^2$ can be classified using elementary arithmetic in a weak arithmetic system, incorporating hypergeometric function results.

Findings

Solutions are classified within a weak arithmetic framework.

Elementary arithmetic suffices for solution classification.

Hypergeometric functions are integrated into the arithmetic framework.

Abstract

The title equations were originally solved by making use of certain results on hypergeometric functions. Aside from these results, the classifications of the solutions uses very elementary arithmetic. The goal of this is to show that these solutions hold in a weak fragment of arithmetic; one strong enough to express the notions of even and odd that has been extended to make use of the results on hypergeometric functions.