Let
An element is called prime, if and only if it's only divisors in S_p are and .
In Apostol's book "An Introduction to Number Theory" I found an exercises, in which one had to show that every number in is either an -prime or a product of -primes.
A number that suffices this property be now called -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 , then there unique with for a yet unspecified
itself has unique representation: with and
Thus one gets the equation:
and can immediately confirm: is complete for any
Now I am interested in all sets , in which all numbers have a unique prime factorization. I would call such a set perfect. However which sets 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.
Easy Math Editor
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:
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
\(
...\)
or\[
...\]
to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
What have you tried?
Is S2 complete? Why, or why not?
Is S3 complete? Why, or why not?
Is S4 complete? Why, or why not?
Is S5 complete? Why, or why not?
The areas that this involves is Modular Arithmetic and related concepts.
Log in to reply
I also got a result now: Sp 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..
Log in to reply
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 :)
You should clarify the definition of " sp prime". I believe what you mean is that "the only divisors of sp that are in Sp are 1 and sp".
For example, 9∈S4, and the only divisors are not 1 and 9 (since it has a divisor of 3).
Log in to reply
Yeah, that edit was necessary.