Alternating factorial

Wikipedia, the free encyclopedia - Cite This Source

In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials.

This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically,

$mathrm\{af\}(n)\; =\; sum\_\{i\; =\; 1\}^n\; (-1)^\{n\; -\; i\}i!$

or with the recurrence relation

$mathrm\{af\}(n)\; =\; n!\; -\; mathrm\{af\}(n\; -\; 1)$

in which af(1) = 1.

The first few alternating factorials are

1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019

For example, the third alternating factorial is 1! − 2! + 3!. The fourth alternating factorial is −1! + 2! - 3! + 4! = 19. Regardless of the parity of n, the last (n^th) summand, n!, is given a positive sign, the (n - 1)^th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly.

This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute values.

Except for n = 1, the factorial of n and the alternating factorial of n are coprime.

Miodrag Zivković proved in 1999 that there are only a finite number of alternating factorials that are also prime numbers, since 3612703 divides af(3612702) and therefore divides af(n) for all n ≥ 3612702. As of 2006, the known primes and probable primes are af(n) for

n = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164

Only the values up to n = 661 have been proved prime in 2006. af(661) is approximately 7.818097272875 × 10¹⁵⁷⁸.

References

• Yves Gallot, Is the number of primes $\{1\; over\; 2\}sum\_\{i\; =\; 0\}^\{n\; -\; 1\}\; i!$ finite?
• Paul Jobling, Guy's problem B43: search for primes of form n!-(n-1)!+(n-2)!-(n-3)!+...+/-1!

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)
This article is licensed under the GNU Free Documentation License.
Last updated on Monday January 08, 2007 at 13:47:41 PST (GMT -0800)
View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation