Sum of Sequence of Alternating Positive and Negative Factorials being Prime

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Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let:

\(\displaystyle m\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!\)
\(\displaystyle \) \(=\) \(\displaystyle n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1\)


The sequence of $n$ such that $m$ is prime begins:

$3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \ldots$

This sequence is A001272 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Let $\map f n$ be defined as:

$\map f n := \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$

First we observe that for $n > 1$:

$\map f n := n! - \map f {n - 1}$


We have:

\(\displaystyle \map f 1\) \(=\) \(\displaystyle 1!\)
\(\displaystyle \) \(=\) \(\displaystyle 1\) which is not prime


\(\displaystyle \map f 2\) \(=\) \(\displaystyle 2! - \map f 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2 - 1\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 1\) which is not prime


\(\displaystyle \map f 3\) \(=\) \(\displaystyle 3! - \map f 2\)
\(\displaystyle \) \(=\) \(\displaystyle 6 - 1\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 5\) which is prime


\(\displaystyle \map f 4\) \(=\) \(\displaystyle 4! - \map f 3\)
\(\displaystyle \) \(=\) \(\displaystyle 24 - 5\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 19\) which is prime


\(\displaystyle \map f 5\) \(=\) \(\displaystyle 5! - \map f 4\)
\(\displaystyle \) \(=\) \(\displaystyle 120 - 19\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 101\) which is prime


\(\displaystyle \map f 6\) \(=\) \(\displaystyle 6! - \map f 5\)
\(\displaystyle \) \(=\) \(\displaystyle 720 - 101\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 619\) which is prime


\(\displaystyle \map f 7\) \(=\) \(\displaystyle 7! - \map f 6\)
\(\displaystyle \) \(=\) \(\displaystyle 5040 - 619\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 4421\) which is prime


\(\displaystyle \map f 8\) \(=\) \(\displaystyle 8! - \map f 7\)
\(\displaystyle \) \(=\) \(\displaystyle 40 \, 320 - 421\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 35 \, 899\) which is prime


\(\displaystyle \map f 9\) \(=\) \(\displaystyle 9! - \map f 8\)
\(\displaystyle \) \(=\) \(\displaystyle 362 \, 880 - 35 \, 899\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 326 \, 981 = 79 \times 4139\) which is not prime


\(\displaystyle \map f {10}\) \(=\) \(\displaystyle 10! - \map f 9\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \, 628 \, 800 - 326 \, 981\) Examples of Factorials
\(\displaystyle \) \(=\) \(\displaystyle 3 \, 301 \, 819\) which is prime


\(\displaystyle \map f {11}\) \(=\) \(\displaystyle 11! - \map f {10}\)
\(\displaystyle \) \(=\) \(\displaystyle 39 \, 916 \, 800 - 3 \, 301 \, 819\) Factorial of $11$
\(\displaystyle \) \(=\) \(\displaystyle 36 \, 614 \, 981 = 13 \times 2 \, 816 \, 537\) which is not prime


\(\displaystyle \map f {12}\) \(=\) \(\displaystyle 12! - \map f {11}\)
\(\displaystyle \) \(=\) \(\displaystyle 479 \, 001 \, 600 - 36 \, 614 \, 981\) Factorial of $12$
\(\displaystyle \) \(=\) \(\displaystyle 442 \, 386 \, 619 = 29 \times 15 \, 254 \, 711\) which is not prime


\(\displaystyle \map f {13}\) \(=\) \(\displaystyle 13! - \map f {12}\)
\(\displaystyle \) \(=\) \(\displaystyle 6 \, 227 \, 020 \, 800 - 36 \, 614 \, 981\) Factorial of $13$
\(\displaystyle \) \(=\) \(\displaystyle 5 \, 784 \, 634 \, 181 = 47 \times 1427 \times 86 \, 249\) which is not prime


\(\displaystyle \map f {14}\) \(=\) \(\displaystyle 14! - \map f {13}\)
\(\displaystyle \) \(=\) \(\displaystyle 87 \, 178 \, 291 \, 200 - 5 \, 784 \, 634 \, 181\) Factorial of $14$
\(\displaystyle \) \(=\) \(\displaystyle 81 \, 393 \, 657 \, 019 = 23 \times 73 \times 211 \times 229 \, 751\) which is not prime


\(\displaystyle \map f {15}\) \(=\) \(\displaystyle 15! - \map f {14}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \, 307 \, 674 \, 368 \, 000 - 81 \, 393 \, 657 \, 019\) Factorial of $15$
\(\displaystyle \) \(=\) \(\displaystyle 1 \, 226 \, 280 \, 710 \, 981\) which is prime


From here on in the numbers become unwieldy.


Sources