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Sun, 17 Jun 2007

Square triangular numbers
A while back I made the erroneous assertion that no numbers are both square and triangular. As I noted in a followup, this is a rather stupid thing to say, since both 0 and 1 are obvious counterexamples. (36 is a nontrivial counterexample.) Also, a few years before I had actually investigated this very question and had determined that the set of such numbers is infinite. Whoops.

I no longer remember how I solved the problem the first time around, but I was tinkering around with it today and came up with an approach that I think is instructive, or at least interesting.

We want to find non-negative integers a and b such that ½(a2 + a) = b2. Or, equivalently, we want a and b such that √(a2 + a) = b√2.

Now, √(a2 + a) is pretty nearly a + ½. So suppose we could find p and q with a + ½ = b·p/q, and p/q a bit larger than √2. a + ½ is a bit too large to be what we want on the left, but p/q is a bit larger than what we want on the right too. Perhaps the fudging on both sides would match up, and we would get √(a2 + a) = b√2 anyway.

If this magic were somehow to occur, then a and b would be the numbers we wanted.

Finding p/q that is a shade over √2 is a well-studied problem, and one of the things I have in my toolbox, because it seems to come up over and over in the solution of other problems, such as this one. It has interesting connections to several other parts of mathematics, and I have written about it here before.

The theoretical part of finding p/q close to √2 is some thing about continued fractions that I don't want to get into today. But the practical part is very simple. The following recurrence generates all the best rational approximations to √2; the farther you carry it, the better the approximation:

p0 = 1 q0 = 1
pi+1 = pi + 2qi qi+1 = pi + qi
This gives us the following examples:

p q p/q
1 1 1.0
3 2 1.5
7 5 1.4
17 12 1.416666666666667
41 29 1.413793103448276
99 70 1.414285714285714
239 169 1.414201183431953
577 408 1.41421568627451
1393 985 1.414213197969543
3363 2378 1.41421362489487
And in all cases p2 - 2q2 = ±1.

Now, we want a + ½ = b·p/q, or equivalently (2a + 1)/2b = p/q. This means we can restrict our attention to the rows of the table that have q even. This is a good thing, because we need p/q a bit larger than √2, and those are precisely the rows with even q. The rows that have q odd have p/q a bit smaller than √2, which is not what we need. So everything is falling into place.

Let's throw away the rows with q odd, put a = (p - 1)/2 and b = q/2, and see what we get:

pqab½(a2+a) = b2
3 2 1 11
17 12 8 636
99 70 49 351225
577 408 288 20441616
3363 2378 1681 11891413721
Lo and behold, our wishful thinking about the fudging on both sides canceling out has come true, and an infinite set of solutions just pops right out.

I have two points to make about this. One is that I have complained in the past about mathematical pedagogy, how the convention is to come up with some magic-seeming guess ahead of time, as when pulling a rabbit from a hat, and then at the end it is revealed to be the right choice, but what really happened was that the author worked out the whole thing, then saw at the end what he would need at the beginning to make it all work, and went back and filled in the details.

That is not what happened here. My apparent luck was real luck. I really didn't know how it was going to come out. I was really just exploring, trying to see if I could get some insight into the answer without necessarily getting all the way there; I thought I might need to go back and do a more careful analysis of the fudge factors, or something. But sometimes when you go exploring you stumble on the destination by accident, and that is what happened this time.

The other point I want to make is that I've written before about how a mixture of equal parts of numerical sloppiness and algebraic tinkering, with a dash of canned theory, can produce useful results, in a sort of alchemical transmutation that turns base metals into gold, or at least silver. Here we see it happen again.


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