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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. 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. 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.

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    You might want to take a look to the OEIS sequence (http://oeis.org/A037153)2011-02-02

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