More Notebook Archeology

I hoped I might finish the exercises from the review of conics today, but I am still feeling under the weather, so I didn’t push that. Instead I read some of The Joy of X by Steven Strogatz and poked around in the same old notebook where I unearthed the problem about superprimes. There I found a nifty, non-inductive proof that $\sum_{i=0}^{n-1}x^i=\frac{x^n-1}{x-1}$, which is a theorem from the first chapter of Number Theory by George E. Andrews, where it is proven by induction.

I also found two unproven propositions about digit sums and the question, “What is the pattern of the sums of digits for multiples of 9?” I worked a bit on the latter, but couldn’t make much of the results. That led me to look them up in the On-Line Encyclopedia of Integer Sequences, a resource I’d heard of but never before used. The sequence I was working with didn’t prove to have any very interesting properties, but I found some others I want to look into further, such as the binary weight of $n$ and the sequence $n$ minus the sum of the digits of $n$, the terms of which are always multiples of 9.¹ (Whoa.) I’ll have to be careful, though. The OEIS looks worse than Wikipedia or even TV Tropes for trapping in the unwary browser.

1. To do. ↩︎