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For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions.

My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer solution,

But for $k=2m+1$, I can't prove three finitely many integral solution.

Thank you.

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This is a result of Størmer's theorem. en.wikipedia.org/wiki/Størmer's_theorem –  Ryan Jun 15 at 4:51
    
Hello,Thank you,can you find somepaper reseacher this problem,I find sometime,I can't find paper. –  math110 Jun 15 at 4:57

1 Answer 1

This is a result of a theorem due to Carl Størmer. His theorem can be stated as follows:

Let $S=\left\{ p_1^{e_1} p_2^{e_2} \cdots p_j^{e_i} \mid e_i \in \mathbb{N}_0 \right\}$, where $p_1$, $p_2 \cdots p_j$ is a finite collection of prime numbers. Then there are a finite number of pairs $x,y \in S$ such that $x-y=k$ (for fixed $k$).

Unfortunately, the wikipedia page for the theorem only considers the $k=1$ case, when in fact Størmer had proven it for all integer values of $k$.

Sources:

  1. http://projecteuclid.org/download/pdf_1/euclid.ijm/1256067456

  2. http://en.wikipedia.org/wiki/Størmer's_theorem

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It's not immediately clear how $k=1$ could have any solutions, since $S$ would be a single-element set... –  abiessu Jun 15 at 5:33
    
Ah, oops, inconsistent variables. This is what happens when I'm running low on sleep.. –  Ryan Jun 15 at 5:36
    
Hello,this theorem how prove it? –  math110 Jun 15 at 6:14
    
@math110 Check the first link, an article titled "On a problem of Stormer" by D. Lehmer. Also I have heard of an article titled "A generalization of the Stormer theorem and some applications" which I have not read, but it sounds promising. These should provide constructive proofs. –  Ryan Jun 15 at 6:26
    
Does stormers theorem guarantee that there exist solutions, or only that there can be at most a finite number of them? –  abiessu Jun 15 at 14:15

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