Understand the problem

Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

Source of the problem

Croatia MO 2005

Topic
Number Theory
Difficulty Level
6/10
Suggested Book
Challenges and Thrills in Pre College Mathematics

Start with hints

Do you really need a hint? Try it first!

Observe that the LHS is sort of a quadratic and the RHS is sort of a linear type. Observe that the given equation is the same as (k!-1)(l! – 1) = m! + 1. Now observe that it implies that m > k,l.
Given that m>l,k, Observe that the equation is symmetric in k and l. WLOG, assume l  \geq k. Now. there are two cases: 1. m > l = k. 2. m > l > k.
1. m > l = k. Observe that it turns out that k! = 2 + (k+1)…m. Hence, k can be 0/1/2/3.  The only solution turns out to be (3,3,4) for (k,l,m).
2. m > l > k. l! = 1 + (k+1)…l + (k+1)…l…m. Hence, it implies that l can only be = 1, but it doesn’t give rise to any solution.  Hence, the only solution is (3,3,4).

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