Possible Natural Numbers

Number Theory Level 3

How many natural numbers $N$ are there such that $N! +10$ is a perfect square ?

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

Infinitely many 1 2 5

1 solution

Brian Charlesworth
Feb 3, 2017

Perfect squares are equivalent to either $0 \pmod{4}$ or $1 \pmod{4}$ .

For $N \ge 4$ we have that $N! + 10 \equiv 2 \pmod{4}$ , so $N \lt 4$ .

Since $1! + 10 = 11, 2! + 10 = 12$ and $3! + 10 = 16 = 4^{2}$ , we can conclude that there is only $\boxed{1}$ natural number $N$ , namely $N = 3$ , such that $N! + 10$ is a perfect square.

Possible Natural Numbers

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