On the Diophantine equation |𝑎𝑥ⁿ-𝑏𝑦ⁿ|=1

Bennett, Michael; de Weger, Benjamin

doi:10.1090/S0025-5718-98-00900-4

On the Diophantine equation $|ax^n-by^n|=1$
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by Michael A. Bennett and Benjamin M. M. de Weger PDF

Math. Comp. 67 (1998), 413-438 Request permission

Abstract:

If $a, b$ and $n$ are positive integers with $b \geq a$ and $n \geq 3$, then the equation of the title possesses at most one solution in positive integers $x$ and $y$, with the possible exceptions of $( a, b, n )$ satisfying $b = a + 1$, $2 \leq a \leq \min \{ 0.3 n, 83 \}$ and $17 \leq n \leq 347$. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.

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