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Suppose that $\{x_n\}$ is an increasing sequence whose elements are positive integers composed of finitely many primes $p_1, \dots, p_s$. I want to verify the following limit $$ \lim_{n\to\infty}x_{n+1}-x_{n}=\infty. $$ I have read a result that gives a lower bound for the difference between the consecutive terms of $\{x_n\}$ in the literature. This result implies that the difference between the consecutive terms diverges. However, can I elementarily show that the limit above is infinite?

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  • $\begingroup$ Related. I'm not sure how elementary Thue's result is. $\endgroup$ Commented Aug 16, 2020 at 12:56
  • $\begingroup$ @DanielFischer, can you see my proof once if it's correct or not? $\endgroup$
    – ShBh
    Commented Aug 16, 2020 at 15:57

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This answer from Felipe Voloch on mathoverflow.net is relevant:

Yes, it is true that this kind of equation ax+by=c, where a,b,c are non-zero and fixed and x,y are allowed to only have prime factors in a finite set, has only finitely many solutions. This is a special case of Siegel's theorem on integral points on curves.

Choose $a=1$ and $b=-1$, so that $x-y=c$ has only finitely many solutions for any given $c$. Therefore there are only finitely many pairs $x,y$ with $|x-y|<M$ for any given $M$.

Unfortunately Siegel's theorem is by no means elementary. I suspect that there is no elementary proof.

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  • $\begingroup$ can you see my proof once if it's correct or not? $\endgroup$
    – ShBh
    Commented Aug 16, 2020 at 15:58

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