More than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, you always end up with , that is, factorial.
Repeating the example I used in that post seven years ago, suppose we choose , and start with the five consecutive integers . We raise them all to the fourth power, giving us
Now we take pairwise differences: , then , and so on, and we get the new list
Repeating the difference procedure gives
And , of course, is .
I came up with a very cool proof of this, and started to explain it, but got stuck on how to explain the Principle of Inclusion-Exclusion (PIE). But now that I’ve finally done that in the last six posts (A probability puzzle, Probabilistic PIE, Have a piece of PIE, Formal PIE, PIE: proof by algebra, PIE: proof by counting), I’m finally going to finish it! Next time I’ll recap the story so far, with links back to my previous posts, and then we’ll forge onward.