On the dynamics of the Fermat-like Diophantine equation involving Pell numbers

In this study, we investigated a Fermat-like Diophantine equation involving Pell numbers, specifically examining powers of two that can be represented as the sum of the $x$-th powers of any two Pell numbers. To solve the equation $P_m^x+P_n^x=2^a$ for $x\ge 3$, where $m$, $n$, $x$, and $a$ are non-negative integers, we employed Baker's theory of linear forms in logarithms, a modified version of the Baker-Davenport reduction method, and properties of continued fractions. Our results extend previous findings for $x=1$ and $x=2$, shedding light on the interplay between Pell numbers and powers of two in this exponential context.