# Requesting assistance regarding unique factorization representation of numbers by s_p primes

Let $S_p := \{np+1 | n \in \mathbb{N_0} \} = \{1, p+1, 2p+1, \dots \}$

An element $s_p \in S_p$ is called $s_p$ prime, if and only if it's only divisors in S_p are $1$ and $s_p$ .

In Apostol's book "An Introduction to Number Theory" I found an exercises, in which one had to show that every number in $S_4$ is either an $s_4$-prime or a product of $s_4$-primes.

A number $p$ that suffices this property be now called $p$-complete. Respectively such a set S_p\ will be called complete. Now one can ask: Which $$p \in \mathbb{N} suffice this property? Well, let $x,y \in S_P$, then there $\exists$ unique $m,n \in \mathbb{N}$ with $k(np+1) = mp+1$ for a yet unspecified $k \in \mathbb{N}$ $k$ itself has unique representation: $k = p*s+t$ with $s \in \mathbb{N}$ and $0 \le t \le p-1$ Thus one gets the equation: $(sp+t)(np+1) = mp+1 \Leftrightarrow spnp +sp+np+t = mp +1 \Leftrightarrow p(nsp +sp+np-m) + t = 1$ and can immediately confirm: $S_p$ is complete for any $p \in \mathbb{N}$ Now I am interested in all sets $S_p$, in which all numbers have a unique prime factorization. I would call such a set $S_p$ perfect. However which sets $S_p$are perfect? How do I tackle this problem? What is a good approach? Any constructive help, recommendation of reading material, comment or answer is appreciated. Thanks in advance. Note by Alisa Meier 4 years, 9 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in \( ... $$ or $ ... $ to ensure proper formatting.
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What have you tried?
Is $S_2$ complete? Why, or why not?
Is $S_3$ complete? Why, or why not?
Is $S_4$ complete? Why, or why not?
Is $S_5$ complete? Why, or why not?

The areas that this involves is Modular Arithmetic and related concepts.

Staff - 4 years, 9 months ago

I also got a result now: $S_p$ is only perfect for p = 1 or p = 2

The proof of that was also not too difficult. Maybe this can be turned into a nice problem for brilliant..

- 4 years, 9 months ago

That's great! It's not too hard, once you figure out the slight trick involved. Looking at small cases can help, which is why I asked.

I look forward to seeing the question that you pose. It could be made really interesting :)

Staff - 4 years, 9 months ago

You should clarify the definition of " $s_p$ prime". I believe what you mean is that "the only divisors of $s_p$ that are in $S_p$ are 1 and $s _ p$".

For example, $9 \in S_ 4$, and the only divisors are not 1 and 9 (since it has a divisor of 3).

Staff - 4 years, 9 months ago

Yeah, that edit was necessary.

- 4 years, 9 months ago