# Factorions Base 10

## Theorem

The following positive integers are the only factorions base $10$:

$1, 2, 145, 40 \, 585$

## Proof

 $\displaystyle 1$ $=$ $\displaystyle 1!$ $\displaystyle 2$ $=$ $\displaystyle 2!$ $\displaystyle 145$ $=$ $\displaystyle 1 + 24 + 120$ $\displaystyle$ $=$ $\displaystyle 1! + 4! + 5!$ $\displaystyle 40 \, 585$ $=$ $\displaystyle 24 + 1 + 120 + 40 \, 320 + 120$ $\displaystyle$ $=$ $\displaystyle 4! + 0! + 5! + 8! + 5!$

## Historical Note

The fact that $40 \, 585$ is a factorion base $10$ was discovered as late as $1964$ by Leigh Janes.