I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but I am looking for results of the equation $ax^n-by^m=1$.
Does $ax^n-by^m=1$ has infinite solution? How do we find them?
Please provide related literature/reference if possible.
2026-04-01 08:54:06.1775033646
Power Diophantine Equation: $ax^n-by^m=1$
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It's a special case of Pillai's conjecture which is a generalization of Catalan's conjecture. Recently, Bernett proved the remarkable result here in 2002 that
Let d be an integer with d > 3 and let a, b be positive with $d \geq 3$ and $a,b$ be positive integers. Then the equation $$|ax^d-by^d|=1$$ has at most 1 solution.
Generally, these equations motivate from Thue's equation.