multifactorial prime
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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)
For example, 7! = 5040, 7!! = 105, 7!!!=28, 7!!!! = 21, and 7!!!!! = 14.  The double factorial notation is sometimes used in statistics texts for combinatorial arguments.

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 for n = 5, 7 , 15 , 17, 19, 51, 73, 89, 131, 153, 245, 333, 441, 463, 825, 1771, and 2027.

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

See Also: Factorial, FactorialPrime, PrimorialPrime

Related pages (outside of this work)

References:

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



Chris K. Caldwell © 1999-2016 (all rights reserved)