I don’t have a factoid about pi, but here’s a math gem that pleases me, about prime numbers. Primes and composite (non-prime) numbers are mixed together throughout the natural numbers, in seeming chaos. Prime numbers can never appear in runs, since even numbers are never prime, but composites clump together. Here’s the question: how many composites can appear consecutively?
It turns out that you can find arbitrarily long runs of composite numbers, there is no limit to how many will appear together. To prove this, here’s a method to find a run of a desired length n. Consider the factorial of n+1, let’s call it Q. Q is 1 × 2 × 3 × ... × (n+1), so Q is divisible by every number from 2 to n+1. Since Q is divisible by 2, Q+2 is divisible by 2. Similarly, Q+3 is divisible by 3, Q+4 is divisible by 4, and so on, all the way up to Q+n+1, which is divisible by n+1. So the run of numbers from Q+2 to Q+n+1 are all composite, and there are n of them in a row, a run of n composites, for whatever n you choose.