Wednesday, September 20, 2006

The magic of √2

Bill Gasarch has an entertaining dialogue on √2 in the latest issue of SIGACT News (which also has an interesting variant of art-gallery problems in the Geometry column). It's a remarkable coincidence, because I was just about to post an entry about a 7th grade classroom problem that revolves around properties of √2). Unlike T, T, F, S, E, this problem actually does have some really nice math behind it.

The problem is as follows:
You live on a street where the houses are numbered 1,2, 3, etc.. You notice that the sum of house numbers prior to yours equals the sum of house numbers after yours. What is your house number, and how many houses are there on the street ? Your answer should be in the 30s
Some quick algebra on n, the number of houses, and k, your house address, reveals that the numbers satisfying this condition yield a square triangular number. Namely, n and k must satisfy the equation

n(n+1)/2 = k2

It turns out that numbers satisfying this equation can be derived from solutions to Pell's equation:

x2 - 2 y2 = 1

where x, y are integers. Pell's equation actually gives good rational approximations to √2, in the form x/y, where x, y are solutions. What's more, if x/y is a convergent in the continued fraction of √2, then x2y2 is a square triangular number (i.e can be written as n(n+1)/2 or k2).

There's also a general form of the solution, that I first found (courtesy David Applegate) in the Hardy/Wright book on number theory. The "trick" here is to realize that the appropriate way to solve this is over the field of numbers of the form a + b√2.

It's rather satisfying that a simple extra credit problem for a 7th grade math class can yield such nuggets.


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