If we are given a prime $p$, naturals $y$ and $N$ such that $p^y$ is approximately equal to $N$, there are some statistics I'm looking for. If we take some of the factors in the prime factorization of $p^y - 1$, how close can we get to $\sqrt{N}$ by forming a product of some of the factors?
Let me give an example. Say $p=3$, $y=5$, and $N = 240$. We have $p^y = 3^5 = 243 \approx N$. Now, we can factor $3^5-1 = 242 = 2 \cdot 11^2$. We can take $2 \cdot 11 = 22$ from these factors, and compared to $\sqrt{240} = 15.4919$ is off by around $7$. This $22$ is our best fit.
Now, we hopefully can create a function that gives the average "best fit" for a particular $p$, $y$, and $N$. What is this function?
Essentially, what I'm looking for is how close we get to $p^{y/2}$, by taking some factors from the prime factorization of $p^y - 1$.
We are allowed to substitute all primes between $q$ and $q^2$ in place of $p$. Hopefully this will give a much better bounds!