Finding all natural number solutions to $2^{\lceil \log_2(3^x) \rceil} = 3^x + 1$

Asked Jan 01 '21 at 09:22

Active Jan 01 '21 at 12:29

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Can someone help on how to find all natural number solutions to this following equation: $$2^{\lceil \log_2(3^x) \rceil} = 3^x + 1$$

Obvious solution is $x = 1$.

Is there any other possible solution? If no other solution, any way we can proof it?

Thanks a lot.

edited Jan 01 '21 at 12:29

Blue

asked Jan 01 '21 at 09:22

Anwar

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Jan 01 '21 at 09:29
https://math.stackexchange.com/questions/164874/2x-3y-1-has-only-three-natural-pairs-as-solutions/164876#164876 – Kenta S Jan 01 '21 at 09:55
The link by Kenta shows all cases where $3^x \pm 1$ is a power of $2$. The only case with $3^x+1$ is your case $x=1$. There are two more cases with $3^x-1$. – GEdgar Jan 01 '21 at 12:56

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