this looks to be an application of the pigeon hole principle.
Powers of 3 Whose Difference Is Divisible by 1997
is a similar problem and solution, scroll down a bit for the solution
Prove that there exist two distinct powers of 3 whose difference is a multiple of 2014. (The exponents need to be non-negative integers. To help clarify the question, if we had asked for 24 instead of 2014, then 3^4 and 3^2 are two powers of 3 whose difference 3^4 - 3^2 = 72 is a multiple of 24.)
My attempt at the solution:
2014|$3^x-3^y$
$3^x \equiv 3^y (mod 2014)$
I don't know where to go from here. How can I solve this for x and y?
this looks to be an application of the pigeon hole principle.
Powers of 3 Whose Difference Is Divisible by 1997
is a similar problem and solution, scroll down a bit for the solution