# Alternating factorial

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

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,

or with the recurrence relation

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 (sequence A005165 in OEIS)

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.

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^{[update]}, the known primes and probable primes are af(*n*) for (sequence A001272 in OEIS)

*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 × 10^{1578}.

## References

- Yves Gallot, Is the number of primes finite?