I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's Elementary Number Theory, but for the LIFE of me I can't figure it out! My initial thought was this: Let m be the smallest prime such that
k<m. Then the answer is
m-2. This is quite clearly not the answer based on the link I provided. My logic was that any number of the form
k!+n will be composite when
1<n<m. That is true. However, it is not the end of the story. Any ideas?
Edit: For anyone curious, this is problem 2 part (b) in section 23.2 of the Second Edition of Underwood Dudley's Elementary Number Theory. Part (a) asks what the smallest integer
n is such that
n+1, n+2, n+3, and
n+4 are all composite.
Second edit: Milcak's comment made me realize that, given some arbitrary
n, we can write
k!+n=k!+i+j so long as
i+j=n. I'm thinking about this now...
Third edit: Here is another problem with the concept for me. I feel like the primes less than or equal to k should help us predict the divisbility of
k!+i, and indeed they do, not when
i is a prime greater than
k. For example,
11!+13=199*200587. That sort of behavior seems unpredictable to me.