Quote:
Originally Posted by JuanTutors
I was trying to find the theorem about the length of the distinct prime factors of a randomly chosen large integer, but I could not find it to estimate the probability of finding a factor.

I don't know about such a theorem, but I do know there is an unproven conjecture called "abc conjecture", which says something about factors of sums. However, I don't it's useful in this case.