# Chance of complete factorization $b^n±1$, with $n$ prime.

I am trying to completely factor numbers of the form $b^n±1$ for some base $b$ $<$ $2^{32}$ and $n$ prime. I am doing this by trial dividing $b^n±1$ to $2^{32}$ and probable prime testing the remaining factor after dividing by all small factors $<$ $2^{32}$. This is for all prime $n$ $<$ $100000$.

Using the method and definition above, what is the chance that I will "completely factor" $b^n±1$ with random prime $n < 100000$ and a certain base $b$ ($±$ sign is fixed with $+1$ or $-1$)?

I find the theoretical value quite complicated for this, does anyone know one? Thanks.

You'll want to look at the Cunningham Project. $(12^{271} - 1)$ is divisible by 1858123207899504867154659615995726699906428378285166639532930145263805746638767329880410679791959, for example.
For $2^n-1$, on June 17, 2015, $n=991$ was completely factored, the last exponent under 1000. More recently, it looks like all the exponents under 1000 for $2^n+1$ have been factored.