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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $19$
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$