Consider the following claim :
(*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then $s_{n+1}-s_n \to \infty$.
If I understand the paper "Exponential Diophantine Equations" by Tidjeman (alvailable here) correctly, then ( * ) is true, was already known to Polya in 1918 and is "an immediate consequence" of Thue’s result on binary forms (this is stated p.382 of the above cited paper). I know about Thue’s theorem, and I am familiar with its applications to polynomial Diophantine equations, but I fail to see its relevance to solving exponential Diophantine equations, so I do not see how (*) can be deduced from it.
Can anyone shed some light on this ?
There's a fairly "dumb" but surprisingly effective trick here: every number of the form $3^n$ is either of the form $x^3$, $3x^3$, or $9x^3$. This idea allows you to map exponential Diophantine equations into finitely many polynomial ones: the new equations are so much less restrictive, but generally better understood.