(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Factorial Equation

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