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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 the 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.

Živković (1999) proved 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.^{[1]} The primes are af(*n*) for

with several higher probable primes that have not been proven prime.

- Weisstein, Eric W. "Alternating Factorial".
*MathWorld*. - Živković, Miodrag (1999). "The number of primes is finite".
*Mathematics of Computation*.**68**(225). American Mathematical Society: 403–409. Bibcode:1999MaCom..68..403Z. doi:10.1090/S0025-5718-99-00990-4. - Yves Gallot, Is the number of primes finite?
- Paul Jobling, Guy's problem B43: search for primes of form n!-(n-1)!+(n-2)!-(n-3)!+...+/-1!