TITLE: Greatest prime factor of n and n+1
QUESTION [15 upvotes]: For a positive integer $n$ we denote its largest prime factor by $\operatorname{gpf}(n)$. Let's call a pair of distinct primes $(p,q)$ $\textbf{nice}$ if there are no natural numbers $n$ such that $\operatorname{gpf}(n)=p, \operatorname{gpf}(n+1)=q$ or $\operatorname{gpf}(n)=q, \operatorname{gpf}(n+1)=p$. For example, $(2,23)$ is nice. 
Are there nice pairs $(p,q)$ with $p,q>100$?

REPLY [8 votes]: So running a simple sieve algorithm allows for recording  pairs which are not nice, and there are many of them below 9 million. I get that the complement includes (2,q) for q=23,29,37,47, and more, (3,q)  for q=89,103,113,131,137 and more, (5,q) for q=307,503,509,613,619 and some more, (7,q) for q=967,971,1031,1039,1049 and some more, (11,q) for q=2381,2543,2551,2591,2801 and a few more, and (13,q) for q=2531,2689,2797.  For larger values of 13 $\lt p \lt q \lt$ 3000, there are no nice pairs.
I am willing to believe there is a q less than 2^2^101 for which (101,q) is nice. Based on preliminary data, I doubt q would be less than 2^101.
Gerhard "These Are Rather Big Numbers" Paseman, 2019.05.31.