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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 ?

Ewan Delanoy
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1 Answers1

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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.

Erick Wong
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  • I've written up more of the details in another thread: http://math.stackexchange.com/questions/724166/is-there-an-integer-y0-such-that-for-all-xy-at-least-4-of-x1-x2-x – Erick Wong Mar 25 '14 at 05:54