Solve Equation: 3^x-2^y=1 - Positive Integer Solutions

[511201]

Can anyone help me with this?
Find all positive integer solutions which satisfies:

(3^x)-(2^y)=1

All I could do (trial and error):
x=1,y=1
x=2,y=4

Is there any formal method this can be solved.?

Thanks

 

[hochs]

now consider x > 1 and y > 1. (Btw, x = 2 and y = 3, not y = 4 in your above list).

mod 3, you find y is odd. mod 4, you find x is even.

So,

3^x - 2^y = 3^x - 2*2^(y-1) = 3, or
(3^(x/2) - sqrt(2) * 2^((y-1)/2))(3^(x/2) + sqrt(2) * 2^((y-1)/2)) = 1, i.e. 3^(x/2) + sqrt(2) * 2^((y-1)/2) is a unit in Z[sqrt(2)], and so

3^(x/2) + sqrt(2) * 2^((y-1)/2) = +/- (1 +/- sqrt(2))^n for some integer n. Now you can derive a contradiction easily.

i.e., use the fundamental units in Z[sqrt(2)].

Remark: The LaTeX support in this website is horrible. I would type it up more neatly if you posted at AOPS.