The Problem
I'm working on a problem assigned in the class Introduction to Abstract Algebra and Number Theory. The question is this:
Consider groups of the form $\left(U_n, *modn\right)$, where $n = 3p$, and $p$ is a prime number. Determine if there is a maximum number of factors of the order of groups of this form.
What I Tried
I know that the order of the multiplicative group of integers modulo n is equal to $\phi\left(n\right)$, so the order of these groups is equal to $2*\left(p-1\right)$.
I was having trouble thinking of ways to approach this problem, so I wrote a Matlab code to find the number of factors of $n=3p$ for all primes less than 10,000. I made a scatter plot for the number of factors of the order of $\left(U_n, *modn\right)$ vs $n$. I can't include it here because my reputation is still too low, but it the highest numbers look to be growing, roughly, logarithmically.
The Question
Can anyone give me some input for how to determine if a set of data is approaching a limit, and how to model that limit if it is?
So the core of your question lies in understanding the number of prime divisors of $p-1$. The fact is that the number of prime divisors of $p-1$ does not have a upper bound while $p$ approaches infinity.
To see this, you may use some constructive approach. For instance, Dirichlet's theorem on arithmetic progressions say that if $A$ and $t$ are relatively prime integers, there are infinitely many primes $q$ of the form $q = Au + t$ with $u \in \mathbb Z$.
To apply to your case, let $A = (\mbox{a product of a whole bunch of primes})$ and $t = 1$, you see that there are infinitely many primes $p$ such that $p = Au + 1$, or $A|(p-1)$.