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:

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


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).


The sequence of those values of $m$ begins:

$5, 19, 101, 619, 4421, 35 \, 899, 3 \, 301 \, 819, 1 \, 226 \, 280 \, 710 \, 981, \ldots$

This sequence is A071828 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Let $\map f n$ be defined as:

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

First we observe that for $n > 1$:

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


We have:

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


From here on in the numbers become unwieldy.


Sources