An alternating factorial prime is an integer sequence prime that is an alternating factorial
 |
that is a prime number. The first few indices 
for which 
is prime or a probable
prime 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). Živković (1999) proved that only finitely
many alternating factorials are prime.
Specifically, he found that the prime number 
divides 
,
so the recurrence relation 
shows that 
divides 
for every 
. Under a heuristic model, he estimated the
total number of prime values to be about 
. The number 
was verified to be prime
in Jul. 2000 by a team of G. La Barbera and others using the Certifix program
developed by Marcel Martin (Jobling 2004).
The following table summarizes the largest known alternating factorial probable primes. Rodenkirch (2017) completed a search through 
showing that none of the additional values through
that limit is a probable prime. Rodenkirch (2026a)
announced plans to sieve through 
. Rodenkirch (2026b) subsequently reported modifying
PFGW to support alternating factorials and
obtaining a sieve bound of 
, with about 425,000 candidates remaining for 
. He planned to verify the implementation
over 
.
 | decimal digits | discoverer |
| 11164 | 40344 | P. Jobling, Nov. 25, 2004 |
| 43592 | 183312 | S. Balatov, Jul. 19, 2017 |
| 59961 | 260448 | M. Rodenkirch, Sep. 18, 2017 |
Is Finite." Math. Comput. 68,
403-409, 1999.