## FANDOM

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The alternating factorial of a number $$n$$ is $$\sum^n_{m = 1} (-1)^{n - m} \cdot m!$$, or the alternating sum of all the factorials up to $$n$$. For example, the alternating factorial of 5 is $$1! - 2! + 3! - 4! + 5!=101$$.[1]

It was Miodrag Živković who proved in 1999 that there are only a finite number of primes that can be expressed as the alternating factorial of a number $$n$$. In particular, the prime 3,612,703 divides all sufficiently large alternating factorial numbers.

## Sources

1. Alternating Factorial -- from Wolfram MathWorld
Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial