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 $$ 3^d +1 = 2^k. $$

See https://en.wikipedia.org/wiki/Catalan%27s_conjecture – lhf Oct 01 '18 at 12:11 — lhf, Oct 01 '18 at 12:11

score 4 · Accepted Answer · answered Oct 01 '18 at 12:08

4

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.

answered Oct 01 '18 at 12:08

asdf

I have forgotten how to manipulate powers of primes modulo powers of other primes. What general results should I look at again to conclude $3^d+1\neq 0 \mod 8$? – Marc Oct 01 '18 at 12:13
$3^k+1$ is congruent to $2$ or $4$ mod $8$. If $k \geq 3$, which is implied by $d \geq 2$, then the LHS is divisible by $8$ which is a contradiction – asdf Oct 01 '18 at 12:27

1 Answers1