2016 and factorial 2

Number Theory Level 4

Find the sum of all the perfect squares that can be expressed as form $\displaystyle \left(\sum_{i=1}^m i! \right)+2016$ .

If you think that no perfect square satisfies this condition, or there are infinitely many perfect squares that satisfy this condition, submit your answer as 999.

Notation : $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

The answer is 2025.

1 solution

Khang Nguyen Thanh
Jun 29, 2016

Suppose that $\left(\displaystyle\sum_{i=1}^m i!\right) +2016=n^2$ with $n$ is a positive integer.

If $m\le 5$ it is easy to check that $m=3$ is the unique solution. In that case, we have: $1!+2!+3!+2016=2025=45^2$

If $m\ge6$ , we have $\left(\displaystyle\sum_{i=1}^m i!\right) +2016\equiv 5\ (\bmod{7})$ , cannot be a perfect square.

2016 and factorial 2

The answer is 2025.

1 solution

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