Consecutive powers (Q792367)

The author gives a very readable history of Catalan's problem. He begins with Levy ben Gerson's solution of Philippe de Vitry's question: Can \(3^ m\pm 1\) be a power of 2 ? The article ends with the announcement of an unpublished result of R. Tijdeman: For a natural number N let S be the set of numbers, all of whose prime factors are less than N. Then there exists a computable constant C, depending only on N, such that if m,n,x,\(y\geq 2\), \(\gcd(ax,k)=1,\quad | a|,\quad | b|,\quad | k| \in S,\) and \(ax^ m-by^ n=k\) then \(\max \{| a|,| b|,| k|,m,n,x,y\}\leq C.\)