Prime factors of a unique form in the each term a sequence?

This can be proven by contradiction. Assume a term has only factors of the form 3n or 3n+1. Then, the term itself must be of the form 3n or 3n+1, which contradicts the fact that all terms in the sequence are of the form 3n-1. Therefore, each term must have at least one prime factor of the form 3n-1. In summary, the conjecture stated in the book can be proven by showing that each term in the sequence must have at least one prime factor of the form 3n-1.
  • #1
musicgold
304
19
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.

Homework Statement


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)##

Homework Equations

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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  • #2
(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.
 
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  • #3
musicgold said:
I am not clear about how I can prove claims ... (3)
It follows directly from (2)
willem2 said:
I really don't know what (2) means.
It is saying that each term must contain a factor which is neither of 3n form nor of 3n+1 form.
 

Related to Prime factors of a unique form in the each term a sequence?

1. What are prime factors?

Prime factors are the numbers that can only be divided by 1 and themselves, without leaving a remainder. For example, the prime factors of 12 are 2, 2, and 3, since 12 can be divided by 2 and 3 without leaving a remainder.

2. What is a unique form in a sequence?

A unique form in a sequence refers to a specific pattern or arrangement of numbers within the sequence that makes it distinct from other sequences. This can be determined by the values, positions, or relationships between the numbers in the sequence.

3. How do you find the prime factors of a unique form in a sequence?

To find the prime factors of a unique form in a sequence, you can start by listing out the numbers in the sequence and then breaking down each number into its prime factors. You can then identify any patterns or relationships between the prime factors to determine the unique form in the sequence.

4. Why is it important to understand prime factors in sequences?

Understanding prime factors in sequences can help us identify patterns and relationships between numbers, which can be useful in solving mathematical problems and predicting future numbers in the sequence. It can also help us identify unique characteristics of different sequences and how they differ from each other.

5. Can prime factors of a unique form in a sequence be used in real-world applications?

Yes, prime factors of a unique form in a sequence can be used in various real-world applications such as cryptography, data compression, and number theory. They can also be used in fields such as finance, biology, and physics to analyze and interpret data.

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