Notation

The notation n! was introduced by Christian Kramp (1760-1826) in 1808 as a convenience to the printer.

In his Élémens d'arithmétique universelle (1808), Kramp wrote:

Je me sers de la notation trés simple n! pour désigner le produit de nombres décroissans depuis n jusqu'à l'unité, savoir n(n - 1)(n - 2) ... 3.2.1. L'emploi continuel de l'analyse combinatoire que je fais dans la plupart de mes démonstrations, a rendu cette notation indispensable.

In his article "Symbols" in the Penny Cyclopaedia (1842) De Morgan complained:

Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results.       ;-))

(From: http://jeff560.tripod.com/stat.html)

Open question: Who started the use of the double exclamatory mark?

Factorians

Some unique Factorials.

        1 = 1!
        2 = 2!
      145 = 1! + 4! + 5!
    40585 = 4! + 0! + 5! + 8! + 5!

These are the only integers with this property. Clifford Pickover calls these numbers Factorians.

Factorial and Power

Let a(n) be the smallest m > 0 such that n^m < m!.
For example 2^4 < 4!, so a(2) = 4.
The sequence a(n) starts 2, 4, 7, 9, 12, 14, ...
This is A065027 at OEIS.

What makes things interesting is the following fact: The differences a(n+1) − a(n) are 2 or 3. Prove it!

Back to the Homepage of Factorial Algorithms.