Are there non-trivial integer solutions for $3^x=2^y+1$?

Question

Consider the equation: $3^x = 2^y+1$

(with $(x,y) \in \mathbb{N}^2$)

There are two easy to find solutions ((1,1) and (2,3)), but it doesn't look like there are more of them.

Are there more solutions to this equation?

Answer 1

Case 1: $y\ge 3$

$3^x = 2^y+1$

$3^x \equiv 1 \pmod 8$

This implies $2|x$ and $\exists\space a$ such that$\space 2a=x$

$3^{2a}=2^y+1$

$3^{2a}-1=2^y$

$(3^a-1)(3^a+1)=2^y$

$\exists \space s,t\in\Bbb{N}$ such that $y=s+t, s>t,\space 3^a+1=2^s,3^a-1=2^t$

$3^a+1=2^s \quad [1]$

$3^a-1=2^t \quad [2]$

subtracting [2] from [1] we have:

$2=2^s-2^t$

$2=2^t(2^{s-t}-1)$

$t=1$ because there is only one factor of $2$ on the left hand side

$2=2(2^{s-1}-1)$

$1=2^{s-1}-1$

$2=2^{s-1}$

$1=s-1$

$2=s$

first solution $y=3,$ $x=2$

case 2: $\quad y=0,$ or$\space y=1,$ or $\space y=2$

by trial and error there is only one solution $y=1,$ $x=1$