If you have n!+(n+1)! and (n+2) is prime, then (n+2) is the largest prime divisor.

The next number, 89, is prime, the largest prime divisor of 87!+88! is 89.

For instance, try 9!+10!, 11 is the largest prime divisor.

29!+30!...the largest prime divisor is 31.

This isn't anything I ever tried to prove. The Hackster probably knows it or will figure it out. There's probably already a known theorem relating it.