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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 For example, the third alternating factorial is 1! + −(2!) + 3! = 5, or if preferred, 1! − 2! + 3! The fourth alternating factorial is −(1!) + 2! + −(3!) + 4! = 19. Regardless of the parity of ''n'', the summand ''n'' − 1 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 Number s. The largest alternating factorial known to be a prime is the alternating factorial of 661, approximately 7.818097272875 × 101578. REFERENCE
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