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| In [[mathematics]], an '''alternating factorial''' is the [[absolute value]] of the [[alternating sum]] of the first ''n'' [[factorial]]s.
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| This is the same as their sum, with the odd-indexed factorials multiplied by [[-1 (number)|−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,
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| :<math>\mathrm{af}(n) = \sum_{i = 1}^n (-1)^{n - i}i!</math>
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| or with the [[recurrence relation]]
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| :<math>\mathrm{af}(n) = n! - \mathrm{af}(n - 1)</math>
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| in which af(1) = 1.
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| The first few alternating factorials are
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| :1, [[1 (number)|1]], [[5 (number)|5]], [[19 (number)|19]], [[101 (number)|101]], 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019 {{OEIS|id=A005165}}
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| 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''<sup>th</sup>) summand, ''n''!, is given a positive sign, the (''n'' - 1)<sup>th</sup> summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly.
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| 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.
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| Miodrag Zivković proved in 1999 that there are only a finite number of alternating factorials that are also [[prime number]]s, since 3612703 divides af(3612702) and therefore divides af(''n'') for all ''n'' ≥ 3612702. {{As of|2006}}, the known primes and [[probable prime]]s are af(''n'') for {{OEIS|id=A001272}}
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| :''n'' = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164
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| Only the values up to ''n'' = 661 have been proved prime in 2006. af(661) is approximately 7.818097272875 × 10<sup>1578</sup>.
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| ==References==
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| * {{MathWorld|urlname=AlternatingFactorial|title=Alternating Factorial}}
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| * Yves Gallot, [http://perso.wanadoo.fr/yves.gallot/papers/lfact.pdf Is the number of primes <math>{1 \over 2}\sum_{i = 0}^{n - 1} i!</math> finite?]
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| * Paul Jobling, [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0411&L=nmbrthry&T=0&P=1106 Guy's problem B43: search for primes of form n!-(n-1)!+(n-2)!-(n-3)!+...+/-1!]
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| [[Category:Integer sequences]]
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| [[Category:Factorial and binomial topics]]
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