HMMT February

Number theory Difficulty 4.8 AIME Find the answer

Find all positive integer solutions $(m, n)$ to the following equation: $m^{2}=1!+2!+\cdots+n!$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

A square must end in the digit $0,1,4,5,6$ , or 9 . If $n \geq 4$ , then $1!+2!+\cdots+n$ ! ends in the digit 3 , so cannot be a square. A simple check for the remaining cases reveals that the only solutions are $(1,1)$ and $(3,3)$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.