It follows that (N-1) consecutive numbers (N! + 2), (N! + 3), ..., (N! + N) are all composite.
N! grows astronomically fast. 10! = 3628800. 70! is larger than the Googol, which is a 1 followed by 100 zeros.
100! is a number with 158 digits. So we are fortunate to have a short and handy notation for this number.
Factorials are indeed used everywhere in Mathematics. Their appearance in the Binomial Theorem and Taylor series
would alone justify a special notation for this product. The word "factorial" is in use from 1800 and the notation N! from 1808. Here is a problem that puts to test your understanding
of this notation.
Problem
Find integer m,n,k such that m!n! = k!.
Solution
Remeber the recursive formula for factorials:
N! = N*(N-1)!?
Well, it's all here. Choose m arbitrarily and let N = m!, n = N-1, and k = N. Written
explicitly, m!*(m! - 1)! = (m!)!.
Reference
- R.Honsberger, More Mathematical Morsels, MAA, New Math Library, 1991
Copyright © 1996-2008 Alexander Bogomolny