# prime factors of a unique form in the each term a sequence?

1. Feb 13, 2019

### musicgold

This is no homework. I have come across a conjecture in a book called The art of the infinite:the pleasures of mathematics. I want to understand how to prove it.

1. The problem statement, all variables and given/known data
Consider a 3-rhythm starting with 2: $2, 5, 8, 11, 14, 17...$
The each number in this sequenc has the form $3n-1$.

The book says that each of these terms can have prime factors of only the following forms: $3n-1,~ 3n,~ 3n+1 .........(1)$
Then it claims that no term could have all factors of the form $3n ~ or 3n+1, ~ or ~ 3n ~ and ~ 3n+1 ........(2)$

Then it claims that each term in the sequence has to have at least one prime factor of the form $3n-1 .......(3)$

2. Relevant equations

3. The attempt at a solution
While I get claim (2) but I am not clear about how I can prove claims (1) and (3). I have tried manually testing and the claims are correct, however I want to prove them. How should I go about it?

2. Feb 14, 2019 at 1:14 AM

### willem2

(1) seems really trivial. All numbers are of the form 3n-1, 3n or 3n+1.
3n isn't possible as prime factor (or any factor) of a number of the form 3n-1. If a number n has a factor that is divisible by 3, n must also be divisible by 3.

I really don't know what (2) means. Factors of the form 3n aren't possible, so it makes no sense to include them here. Is it meant that no number of the form 3n-1 can have only prime factors of the form 3n+1 ? If you can prove that, than (3) would immediately follow.

Last edited: Feb 14, 2019 at 1:24 AM
3. Feb 14, 2019 at 7:07 PM

### haruspex

It follows directly from (2)
It is saying that each term must contain a factor which is neither of 3n form nor of 3n+1 form.