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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.

The first few values n for which  are (probable) primes are 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100).

## 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
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