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Lower bounds for Catalan’s equation. (Minorations pour l’équation de Catalan.) (French) Zbl 0887.11018

The authors prove the following theorem for Catalan’s equation \(x^p-y^q = 1\) \((p,q\) primes \(|x|, |y|>1)\): Besides the obvious solution \(3^2-2^3=1\), any other solution must satisfy \(\min (p,q)> 10^5\) and \(\max (p,q) >10^6\).
The proof is based on two algebraic criteria; one is due to W. Schwarz and is an improvement of an older criterion of Inkeri, and the other is due to the first named author. For the application of these criteria, very heavy computer calculations are required.
Another interesting result based on the above theorem is the following corollary: Let \(p,q\) be as above and \(q\equiv 3\mod 4\), \(q>p\). Suppose that Catalan’s equation has a nontrivial solution for these values of \(p,q\). If \(p\equiv 3 \pmod 4\) then \(p^{q-1} \equiv 1\pmod {q^2}\) and \(q^{p-1} \equiv 1\pmod {p^2}\). If \(p\equiv 1 \pmod 4\) then \(q^{p-1} \equiv 1\pmod {p^2}\).

MSC:

11D61 Exponential Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
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