Small two-variable exponential Diophantine equations

Styer, Robert

doi:10.1090/S0025-5718-1993-1160277-9

Small two-variable exponential Diophantine equations
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Math. Comp. 60 (1993), 811-816 Request permission

Abstract:

We examine exponential Diophantine equations of the form $a{b^x} = c{d^y} + e$. Consider $a \leq 50$, $c \leq 50$, $|e|\; \leq 1000$, and b and d from the set of primes 2, 3, 5, 7, 11, and 13. Our work proves that no equation with parameters in these ranges can have solutions with $x > 18$. Our algorithm formalizes and extends a method used by Guy, Lacampagne, and Selfridge in 1987.

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On the equation

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