1. The problem statement, all variables and given/known data Find all positive integer solutions for a, b and c to the equation: a!b! = a! + b! + c! 2. Relevant equations n! = n(n-1)(n-2)... 3. The attempt at a solution I'm not having much progress with this. I've tried rewriting it as (a! - 1)(b! - 1) = c! + 1 and I've found one solution by observation (a = 3, b = 3, c = 4), but I'm not sure what to do. I've ruled out the case a = b = c (since putting that in gives you a! = 3, which has no integer solutions). I've tried comparing the sets of a and b and have tried pairing up numbers by multiplying them to give an element in the set of possible values of c, but this has not worked either, and given that a calculator is not allowed, it would take too long to compute all the possibilities. A hint (but not a complete solution) would be appreciated here. EDIT: I've noticed that the remaining solutions all involve a, b and c ≥ 5 (I think). I remembered that, for n > 5 is composite iff (n-1)! = 0 mod n, but I'm not sure how to apply this here?