# 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:

\(\ds 0!! + 1\) | \(=\) | \(\ds 1 + 1\) | Definition of Double Factorial | |||||||||||

\(\ds \) | \(=\) | \(\ds 2\) | which is prime |

\(\ds 1!! + 1\) | \(=\) | \(\ds 1 + 1\) | Definition of Double Factorial | |||||||||||

\(\ds \) | \(=\) | \(\ds 2\) | which is prime |

\(\ds 2!! + 1\) | \(=\) | \(\ds 2 \times 0!! + 1\) | Definition of Double Factorial | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 1 + 1\) | Definition of Double Factorial | |||||||||||

\(\ds \) | \(=\) | \(\ds 3\) | which is prime |

This theorem requires a proof.In particular: It remains to be shown that the other integers are the only other ones with this property. Have fun.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1993: Chris Caldwell and Harvey Dubner:
*Primorial, Factorial and Multifactorial Primes*(*Mathematical Spectrum***Vol. 26**,*no. 1*: pp. 1 – 7) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $518$