# Prime Values of Double Factorial plus 1

## Theorem

Let $n!!$ denote the double factorial function.

The sequence of positive integers $n$ such that $n!! + 1$ is prime begins:

$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$

## Proof

We have that:

 $\displaystyle 0!! + 1$ $=$ $\displaystyle 1 + 1$ Definition of Double Factorial $\displaystyle$ $=$ $\displaystyle 2$ which is prime

 $\displaystyle 1!! + 1$ $=$ $\displaystyle 1 + 1$ Definition of Double Factorial $\displaystyle$ $=$ $\displaystyle 2$ which is prime

 $\displaystyle 2!! + 1$ $=$ $\displaystyle 2 \times 0!! + 1$ Definition of Double Factorial $\displaystyle$ $=$ $\displaystyle 2 \times 1 + 1$ Definition of Double Factorial $\displaystyle$ $=$ $\displaystyle 3$ which is prime

## Historical Note

According to 1997: David Wells: Curious and Interesting Numbers (2nd ed.), this result is reported in Volume $26$ of Mathematical Spectrum, but it has not proved possible to confirm this by checking it directly.