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# multifactorial prime

It seems natural (to some authors) to generalize the notion of factorial primes by using the multifactorial functions:*n*! = (*n*)(*n*-1)(*n*-2)...(1)*n*!! = (*n*)(*n*-2)(*n*-4)...(1 or 2)*n*!!! = (*n*)(*n*-3)(*n*-6)...(1,2 or 3)

Since 7!!!!! can be hard to read (those of us getting older lose count of the !'s) and is easy to confuse with
the huge number (((((7!)!)!)!)!, we will write 7!_{5} (i.e. using 5 as a subscript to the exclamation mark). More generally,

*n*!_{j}=*n*if 0__<__*n*__<__*j*, and*n*!_{j}=*n*^{.}(*n*-*j*)!_{j}

**Multifactorial primes**are primes of the forms

*n*!!+/-1,

*n*!!!+/-1,

*n*!!!!+/-1, and so on.

Ken Davis suggests that we also consider the forms
*n*!!± 2. Checking to *n*=5000 he has
found:

n!!-2 is prime forn= 5, 7 , 15 , 17, 19, 51, 73, 89, 131, 153, 245, 333, 441, 463, 825, 1771, and 2027.

n!!+2 is prime forn= 3, 5 ,7 ,9, 21, 23, 27, 57, 75, 103, 169, 219, 245, 461, 695, 1169; and probable-prime forn=3597, 3637.

**See Also:** Factorial, FactorialPrime, PrimorialPrime

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

**References:**

- CD93
C. CaldwellandH. Dubner, "Primorial, factorial and multifactorial primes,"Math. Spectrum,26:1 (1993/4) 1--7.

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.