Determine the squared-parameter relation for Gamma

Determine a convenient way to determine \(\Gamma(a^2,b^2)\) from \(\Gamma(a,b)\) in general, extending the explicitly established cases \(b=a+1\) and \(b=a+2\).

Background

The paper studies the function $\Gamma(a,b)$, which records which of two related Diophantine equations involving coprime parameters $a$ and $b$ has a nonnegative integral solution. The authors investigate how this function behaves when both parameters are squared, motivated by a problem attributed to Chu, Miller, and Tresch asking for a relation between $\Gamma(a,b)$ and $\Gamma(a^2,b^2)$.

The paper proves explicit results for the neighboring-parameter cases $b=a+1$ and $b=a+2$, including parity- and congruence-dependent formulas. However, it does not provide a general method for recovering $\Gamma(a^2,b^2)$ from $\Gamma(a,b)$, leaving the broader relation unresolved.