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# factorial prime

The only factorial that is prime is 2!, so if "factorial primes" are to be worth mentioning, the term must mean something other than a factorial that is prime. In fact, as usually defined,**factorial primes**come in two flavors: factorials plus one (

*n*!+1) and factorials minus one (

*n*!-1). It is conjectured that there are infinitely many of each of these.

*n*!+1 is prime for*n*=1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, and 26951 (107707 digits).*n*!-1 is prime for*n*=3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, and 34790 (142891 digits).

*n*=37000 [CG2000].

**See Also:** Factorial, PrimorialPrime, MultifactorialPrime

**Related pages** (outside of this work)

- Organized searches for factorial primes (check status or perhaps join in!)
- Primorial Primes The top twenty
- Factorial Primes The top twenty
- Deficient factorials by Rene Dohmen

**References:**

- BCP82
J. P. Buhler,R. E. CrandallandM. A. Penk, "Primes of the formn!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,38:158 (1982) 639--643. Corrigendum inMath. Comp.40(1983), 727.MR 83c:10006- Borning72
A. Borning, "Some results fork!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,26(1972) 567--570.MR 46:7133- Caldwell95
C. Caldwell, "On the primality ofn!± 1 and 2 · 3 · 5^{...}p± 1,"Math. Comp.,64:2 (1995) 889--890.MR 95g:11003- CG2000
C. CaldwellandY. Gallot, "On the primality ofn!± 1 and 2 × 3 × 5 ×^{...}×p± 1,"Math. Comp.,71:237 (2002) 441--448.MR 2002g:11011(Abstract available) (Annotation available)- Templer80
M. Templer, "On the primality ofk! +1 and 2*3*5*^{...}* p +1,"Math. Comp.,34(1980) 303-304.MR 80j:10010

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