On the Catalan equation. II
Consider Catalan's equation \(x^ p - y^ q = 1\), where \(p,q \geq 2\) and \(| x |\), \(| y |>1\). Here \(p\) and \(q\) are chosen to be odd primes with \(p<q\). The author continues his investigations (cf. Sur l'équation de Catalan, [C. R. Acad. Sci., Paris, Sér. I 314, 165-168 (1992; Zbl 0749.11025)]) by introducing a new linear form in logarithms which enables him to improve the general upper bounds on \(p\) and \(q\), in case \(p \equiv 3 \pmod 4\). Moreover, it is shown that Catalan's equation has no solution when \(p = 19\) and that for \(p = 53\), the only possible value for \(q\) is \(q = 4889\).
scientific article; zbMATH DE number 501214