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 in $S_p $divisors 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 a 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.
You have unique factorization only for $p=1$ and $p=2$. That you have unique factorization in those cases is easy to see (once these are all natural numbers once these are all odd natural numbers).
Otherwise there are two distinct (usual) primes $q,q'$ such that $q=-1 + np $ and $q'=-1 + n'p$. Then $q,q' $ are not in $S_p$. Yet $q^2 , q'^2 , qq'$ are in $S_p$ and have no divisors there and are thus $s_p$ prime. Thus $(qq')^2$ can be factored as $(qq') \ (qq')$ and $q^2 \ q'^2$.
There is a theory around such questions, see for example "Arithmetic Congruence Monoids: A Survey" by Baginski and Chapman. In: Combinatorial and Additive Number Theory, Volume 101 of the series Springer Proceedings in Mathematics & Statistics, pp 15-38. (There should be a free preprint on the web, too.)