# Difference between the consecutive terms of an increasing sequence consisting of positive integers composed of finitely many primes

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?

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

## 1 Answer

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 $$|x-y| for any given $$M$$.

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

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