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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.

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References

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!