Number Theory in Japan

Lake Ashi, Hakone, Japan

A few weeks ago I was in Hakone, Japan, in the shadow of Mt. Fuji, famous for its hot springs. I was playing cards with some backpackers when an American couple in their 20s, Kevin and Jenny, came in. We asked if they wanted to join us. They demurred. I asked what they did. Kevin turned out to be a programer, board game designer, and former pro-StarCraft gamer. Jenny was working on a math Ph.D. at Carnegie Mellon.​

So they were huge nerds, but so am I. In college I originally meant to study math and physics and become a physicist. I mentioned this to Jenny as the card game wound down and Kevin and I began to play Go, the ancient chess (except far more complex than chess) game that Confucius used to play. Then somehow I mentioned that many of the writers on The Simpsons were mathematicians and wrote many a math joke into the show. “There was one, for example,” I said, “about taxicab numbers.” (Actually I was wrong about this - it was Futurama.)

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Homer Simpson nearly disproves Fermat's Last Theorem. Fair Use.

Jenny’s brows went up. “1729,” she said.

Taxicab numbers are numbers that are the sums of perfect cubes in two different ways, like so: 1729 = 1^3 + 12^3 = 9^3 + 10^3. They’re named after the taxi that British mathematician G.H. Hardy took to see Srinavasa Ramanujan, the real-life Will Hunting, which happened to be cab number 1729, which happens to be the smallest taxicab number in existence.

I told Jenny that I had participated in the Putnam math competition in freshman year of college, when I solved one problem out of ten and was never more proud of myself. See, the Putnam is so difficult that most students who do it end up with a grand total score of zero. Students who come in the top five are automatically accepted to Ph.D. programs at Harvard, and many, including my childhood hero Richard Feynman, go on to win Nobel prizes or Fields Medals.

“How many did you do?” I asked, assuming her participation.

“Four or five,” she said.

In my feeling of inferiority I proceeded to recall the one problem I did solve, so proud I was that it was seared in my memory: “Prove that there exists an infinite number of positive integers n such that n is the sum of two perfect squares, n+1 is the sum of two perfect squares, and n+2 is also the sum of two perfect squares.”

Of course then Jenny took a pen and paper and started working on it. I pointed out that the problem took me five hours to figure out. But true to form she would not be deterred. Twenty minutes later she announced that she’d gotten it, probably, because of something about Diophantine Equations. 

I knew nothing about Diophantine Equations, and still don't even though I Googled them later. But presumably had I known about them in college, the one problem I solved would have taken a lot less time.

I remembered my encounter with Jenny and Kevin some days later as I realized that they had rekindled my interest in mathematics just so much. Of course I’m still not going to get back into it — there was a reason I wound up going to law school. But how nice it was to be reminded of the intellectual pursuits of one’s first youth.