Find all solution of the following equation : $3^{m}-2^{n}=1$

Question

problem :

Find all solution of the following equation :

$$3^{m}-2^{n}=1$$

Where $m,n\in\mathbb{N}$

My try but not complete :

In first we see that $m=n=1$ is a solution

Also: $m=2,n=3$ a solution But I don't know if have another solution or no

Is this equation has many solution ?

$m=1,n=1$? But $m=n=0 \implies 3^m-2^n = 1-1 = 0$ – hola Dec 26 '19 at 15:54 — hola, Dec 26 '19 at 15:54
Yes sir thank you very much – Ellen Ellen Dec 26 '19 at 16:00 — Ellen Ellen, Dec 26 '19 at 16:00

score 0 · Answer 1 · answered Dec 26 '19 at 15:58

0

This is a special case of Catalan's conjecture. The only solution is $m=2, n=3$

answered Dec 26 '19 at 15:58

Zerox

I see but how we prove that ? 2,3 are only a solution ? – Ellen Ellen Dec 26 '19 at 16:06
Welcome! Would you mind giving a link/tutorial (instead of just a name)? – hola Dec 26 '19 at 16:16
https://en.wikipedia.org/wiki/Catalan%27s_conjecture if you don't mind. – Zerox Dec 26 '19 at 18:34

1 Answers1