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
Main article: Factorial
Multifactorials: Double factorial · MultifactorialFalling 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
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial