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

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.