Powers of 3 modulo powers of 2

Question

This question is probably simple, but it has been a couple of years since I have practised group theory.

Let $d\ge 2$ be an integer. Does there exist an integer $k\in\mathbb{N}$ such that $$ 3^d +1 = 2^k. $$

Answer 1

I don't know how this is related to group theory, but you can look at the equation $\mod 8$ and see immediately that there are no solutions.