Prove a number of form $3^n - 1$ can never be a prime

(n 1 is an integer)

Question

Prove a number of form $3^n - 1$ can never be a prime

(n > 1 is an integer)

ganeshie8 · Answer

@Oleg3321

ganeshie8 · Answer

any ideas ?

ganeshie8 · Answer

here is a nice property 
every number of form $3^n-1$ is divisible by $2$; that is, an even number

oleg3321 · Answer

ohhhhhh, i see now, so simple lol :)

ganeshie8 · Answer

How do you know every number of that form is divisible by 2 ?

oleg3321 · Answer

Well when you put any number is the degree of 3 the number will be odd but then you subtract 1 and that makes the number even, which makes it non-prime, correct?

ganeshie8 · Answer

That's really clever!
I had never thought of it that way. great job !

welshfella · Answer

this problem is deceptively simple.

mathmate · Answer

:)

ganeshie8 · Answer

Indeed! Using @Oleg3321 's logic, we can say $a^n−1$ is never prime for all odd numbers $a$

kainui · Answer

Just cause I like it,

$$\frac{3^n-1}{3-1} = \sum_{k=0}^{n-1} 3^k$$
So you can write:

$$3^n-1 = \sum_{k=0}^{n-1}2*3^k$$
which means in base 3, this number looks like:
$$(3^n-1)_3=\underset{\text{n times}}{\underbrace{22\cdots22}}$$

Just like 100-1=99 same thing. Cute observation I wanted to share. Sometimes you can factor numbers this way easily like this:
$$222222 = 2*111111=2*1001*111=2*10101*11$$

ganeshie8 · Answer

Awesomeness! So that basically works for every integer a>2. Just need to look at the number in base 'a'

(a^n - 1)_a = aaa... n times