Patrick Vennebush of Math Jokes 4 Mathy Folks recently wrote about the following procedure that yields surprising results. Choose some positive integer . Now, starting with consecutive integers, raise each integer to the th power. Then take pairwise differences by subtracting the first number from the second, the second from the third, and so on, resulting in a list of only numbers. Do the same thing again, resulting in numbers, and repeat until you are left with a single number.
For example, 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
OK, so we get . So what?
Well, if you try enough examples, you may notice a surprising pattern. I’ll let you play with it for a while. Over the course of a few future posts I’ll explain the pattern and prove that it always holds—but the proof will be a really cool combinatorial one, with pretty pictures!