Alternating Factorial

The alternating factorial is defined as the sum of consecutive factorials with alternating signs,

(1)

They can be given in closed form as

(2)

where  is the exponential integral,  is the En-function, and  is the gamma function.

The alternating factorial will is implemented in the Wolfram Language as AlternatingFactorial[n].

A simple recurrence equation for  is given by

(3)

where .

For , 2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).

The first few values  for which  are (probable) primes are , 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). Živković (1999) has shown that the number of such primes is finite.  was verified to be prime in Jul. 2000 by team of G. La Barbera and others using the Certifix program developed by Marcel Martin.

The following table summarizes the largest known alternating factorial probable primes. M. Rodenkirch completed a search up  in December 2017 showing there are no further (probable) primes up to that limit.

	decimal digits	discoverer
11164	40344	P. Jobbing, Nov. 25, 2004
43592	183312	S. Balatov, Jul. 19, 2017
59961	260448	M. Rodenkirch, Sep. 18, 2017

REFERENCES:

Balatov, S. "Alternating Factorials." Jul. 19, 2017. http://www.mersenneforum.org/showpost.php?p=463778&postcount=7.

Guy, R. K. "Equal Products of Factorials," "Alternating Sums of Factorials," and "Equations Involving Factorial ." §B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193-194, 1994.

Jobling, P. "Guy's Problem B43: Search for Primes of Form n!-(n-1)!+(n-2)!-(n-3)!+...+/-1!." 25 Nov 2004. http://listserv.nodak.edu/scripts/wa.exe?A1=ind0411&L=nmbrthry#4.

Rodenkirch, M. "Alternating Factorials." Dec. 15, 2017. http://www.mersenneforum.org/showthread.php?p=474083#post474083.

Sloane, N. J. A. Sequences A001272, A005165/M3892 in "The On-Line Encyclopedia of Integer Sequences."

Živković, M. "The Number of Primes  is Finite." Math. Comput. 68, 403-409, 1999.