This is no homework. I have come across a conjecture in a book called(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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

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