Wikipedia:Reference desk/Archives/Mathematics/2021 November 4

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November 4

｜3^x − 2^y｜ = 1

Does the equation $|3^{x}-2^{y}|=1$, have any solutions in natural numbers, besides the three obvious ones:

• $3^{1}-2^{1}=1$.
• $2^{2}-3^{1}=1$.
• $3^{2}-2^{3}=1$.

My conjecture is that this equation has no other solutions in natural numbers. It turns out to be true, for all natural exponentials not larger than 10,000. However. for larger natural exponentials, the calculation becomes too slow, so I haven't checked out any larger natural exponentials. 185.24.76.181 (talk) 17:29, 4 November 2021 (UTC)

I don't know the answer, but just a heads-up that it took me a second to figure out that you were asking for solutions in the natural numbers. A "natural solution" could be just one that comes up naturally. --Trovatore (talk) 18:06, 4 November 2021 (UTC)

I've just rephrased my original post in accordance with your comment. Thanks. 185.24.76.181 (talk) 18:12, 4 November 2021 (UTC)

Theoretically speaking, I see no reason why the conjecture should be true. But of course if it is true then it needs a proof. --2A02:AA1:1625:94FC:B128:EC4E:46A0:EC31 (talk) 19:39, 4 November 2021 (UTC)

I believe this is Catalan's conjecture, which, according to our article, is no longer a conjecture as of 2002. --RDBury (talk) 19:54, 4 November 2021 (UTC)

Thanks. 185.24.76.181 (talk) 20:11, 4 November 2021 (UTC)

As can be found in the section Catalan's conjecture § History, this special case was proved already in 1343 by Gersonides. It can be proved by a rather elementary argument.  --Lambiam 20:35, 4 November 2021 (UTC)

Thanks. 185.24.76.181 (talk) 13:55, 5 November 2021 (UTC)