Prime Values of Double Factorial plus 1

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Theorem

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

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

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

This sequence is A080778 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


Sources