Today I relistened to more of A Brief History of Mathematics. That might seem like it hardly qualifies as study, but I think it is helping revive my desire to continue this project. It comprises mathematical bedtime stories, told by an enthusiast, with a very light sprinkling of actual mathematical ideas—just enough to be tantalizing without being overwhelming.
I also thought a bit about one of those ideas, a theorem in number theory discovered by Gauss: that every natural number is the sum of three triangular numbers. I didn’t put my full focus into it, though, and I didn’t come up with any insights.
[Later note: The theorem only holds as stated if we let 0 be a triangular number. Definitions appear to differ on that point.]
I must have met that theorem about triangular numbers sometime before, but I’ve forgotten it. It seems quite remarkable to me.
Triangular numbers, $n(n+1)/2$, are a lot like squares. They have squares in them, and they are roughly twice as abundant as squares. It’s common knowledge among those for whom it ìs common knowledge that every natural number is a sum of 4 squares. I would therefore have guessed that every natural number was a sum of 4 triangular numbers. What’s special about triangular numbers that makes it take only 3 of them? To write the number 7, you need 4 squares, $7=4+1+1+1$, but only 3 triangular numbers, $7=6+1+1$.
This blog is great both because I care about you and because I learn cool math here.
I still haven’t sat down to see if I can make anything of this theorem. I may decide to look up the proof, since as you point out below, it is probably a tricky one.
Speaking of tricky things, further research leads me to believe that strictly speaking, 7 may be $6+1,$ not $6+1+1,$ as claimed above. I’m continuing to investigate this question.
And now I learn that there’s an amazing and more general result called Fermat’s Polygonal Number Theorem: every positive integer is a sum of 3 triangular numbers or 4 squares or 5 pentagonal numbers or 6 hexagonal numbers or… All these numbers are roughly equally abundant, so why is this the way it works?
These results can’t be completely trivial, if finding the proof of the Triangular Number Theorem made Gauss write in his journal “ΕΥΡΗΚΑ! num = Δ + Δ + Δ”. (That’s EUREKA, for those who aren’t as cool as Gauss, and who therefore wouldn’t have written it in Greek.)
Thanks again, Olly. You’re great!
Wow. That is extremely cool.
And very few of us are as cool as Gauss.
I admire your persistence.
Thanks.