The Equations $2^n \pm 2^m \pm 1 = x^2$ in the Arithmetic of the Even and Odd
Matt Wicks
TL;DR
The paper investigates the minimal logical strength required to prove that the Diophantine equations $2^n \pm 2^m \pm 1 = x^2$ have solutions by embedding the problem in a weak arithmetic, $\mathcal{AOE}$, extended to $\mathcal{AOE+B}$ to handle division by powers of two via $\left[ \frac{x}{2} \right]$ and a power decomposition $n=\tau(n)\cdot\omega(n)$ with $\tau(n)$ a power of two and $\omega(n)$ odd. It systematically translates Beukers’ hypergeometric results into the $\mathcal{AOE+B}$ language and uses them to realize Szalay’s theorems, delivering a finite, explicit classification of solutions for the relevant equations and identifying the necessity of additional axioms for a complete translation of one theorem. The work demonstrates that such elementary Diophantine classifications can be achieved within a carefully extended but still finite axiom system, complemented by bounded computational checks. Overall, it clarifies the logical resources required to establish these results and shows how hypergeometric techniques can be reformulated in weak arithmetic settings.
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.