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Home / Opinion / Columns /  Fifteen at one blow, take the escalator

There is such a thing in mathematics as the "Fifteen Theorem". Not the "fifteenth theorem" out of the thousands upon thousands that are out there. I mean the "Fifteen Theorem".

The name itself is intriguing. Mathematics has many famous theorems: the Pythagoras Theorem, Gödel's Incompleteness Theorem, Fermat's Last Theorem and more. There are others that have a number in the name - Four Squares Theorem, Four Colour Theorem ... but is there one named for a number? I don't think so.

Because the name is so intriguing, this column will try to explore the Fifteen Theorem.

It was first stated in 1993 by the late great John Conway and WA Schneeberger, and here it is:

The Fifteen Theorem: If a positive-definite quadratic form having integer matrix represents every positive integer up to 15, then it represents every positive integer.

I spent a few minutes staring at those words. Apart from the expected mention of 15, they reminded me of the elegant proof technique of induction, which I first ran into it in early college-level mathematics. In essence: if something you want to prove is true for a particular "base case", and if you can move from there to the next case (the "inductive step"), this much completes the proof.

If that seems translucent at best, consider two examples as metaphors.

First, climbing a flight of stairs. How will you persuade me that I can climb the whole flight? First, by showing that I can set foot on the first stair. Second, by showing that from there, I can step up onto the second stair. That is all I need. Because now that I'm perched on the second stair, I use the same step up to reach the third, then the fourth, fifth and so on ... and suddenly I've reached the top.

Second, breaking a bar of chocolate. Let's say the bar is divided by the usual shallow lines into 20 smaller squares. How many breaks do I need to get 20 individual pieces? Well, start with a bar consisting of one square. Zero breaks needed. Two squares? Clearly, just one break. Three squares? One break gives you an individual piece and a two-square bar, and the latter needs one more break. So a three-square bar needs two breaks - and so on. With each break, we reduce a bar to two earlier cases (e.g. single piece and two-square bar), whose count of breaks we know. Thus we figure: a 20-square bar needs 19 breaks.

Metaphors like these help make clear what proof by induction really means. Again, it needs two components: a "base case" (set foot on step #1, one-square chocolate bar) and an "inductive step" (climb up to the next step, break off a chocolate square). In mathematics, induction is a powerful tool in the proof of innumerable theorems. In the statement of the Fifteen Theorem, to me there's a hint of induction. What it says is that if we can prove something complicated (whatever it is) for integers up to 15 - there's the base case - then we have proved it for all integers.

But what is that complicated thing?

Let me try to get there by starting with the Four-Squares Theorem that made an appearance above. This dates from 1770, when Joseph-Louis Lagrange proved a fascinating proposition indeed. He showed that any natural number (that is, 0 and the positive integers 1, 2, 3 and so on) can be expressed as the sum of no more than four squares of the natural numbers (the numbers 0, 1, 4, 9, 16 and so on).

Try this for yourself:

0 = 0

45 = 36 + 9

69 = 64 + 4 + 1

142 = 121 + 16 + 4 + 1

etc.

In algebraic terms, Lagrange proved that any given natural number N can be written like this:

N = a2 + b2 + c2 + d2

in which a, b, c and d are also natural numbers. Taking the fourth example above, N is 142; a is 11, b is 4, c is 2 and d is 1. Since this is true for any natural number N, mathematicians say this equation is "universal".

Obsessed with numbers that we humans are, we've been fiddling with these kinds of manipulations for a very long time. For example, why just this addition of four squares? What if we multiplied one of them by, say, 3; another by 5; and a third by 14? In other words, what if we give a, b, c and d coefficients? In this case, if we wrote this:

N = a2 + 14b2 + 3c2 + 5d2

would it still hold for any natural numbers N, a, b, c and d - that is, is this equation also universal? Well, the number theorist genius Srinivasa Ramanujan wondered, among a whole lot else, about questions like this. It should be no surprise that he proved that there are 55 equations like these that are universal, though later, he was shown to be wrong about one of those. As it turns out, the one just above is not in his list. But many others are, like:

N = a2 + 12b2 + 4c2 + 2d2

N = 7a2 + b2 + 2c2 + 3d2

We are getting closer to the Fifteen Theorem, I promise you. When he was teaching at Princeton University in 1993, John Conway and his students worked on equations like these. Not just Ramanujan's 54, but all such equations that use any coordinates at all for a, b, c and d: were they all universal? My intuition is that Conway had induction on his mind, because he thought, why not check a few values for N? If we can show that any such equation can produce those values - a base case, if you like - maybe that will lead to a proof of universality.

Conway was right. They were able to show that if the equation produces the numbers up to 15 (1, 2, 3, 4 ... 15), it is universal.

There is something here that leaves me almost speechless. Think of it. Take a random equation like the ones above:

N = 131a2 + 3b2 + 16c2 + 8d2

Conway proved that if this can generate 1, 2, 3 ... all the way to 15 (just 15!), it will also generate 971,402 and 7,444,333,131,229 and in fact, every other natural number.

Now you know why it's called the Fifteen Theorem. You may also have figured that "positive-definite quadratic form having integer matrix" - taken from the statement of the theorem above - is just math-speak for these equations we've been discussing. The proof that Conway and the Princeton students found was complicated and they never published it. But in 2000, and fittingly because this came down from Ramanujan, the stellar Canadian mathematician of Indian origin, Manjul Bhargava, found a much simpler proof (see page 27 ).

Simple it might be for mathematicians, but it is way beyond my mathematical ken. Still, there's a sweet reminder of the metaphor I used above about stairs. Bhargava uses not stairs, but a numerical device he calls "escalator lattices". He speaks of "escalating" two-dimensional escalators to three dimensions, then four and finally five dimensions. Never mind what all this means. But "with a bit of fear," writes Bhargava, "we may ask again whether any of these five-dimensional escalators escalate." That is, are we going to escalate forever?

"Fortunately, the answer is no," Bhargava tells us. "All five-dimensional escalators are universal."

And that proves the Fifteen Theorem. We're at the top of the staircase. Have some chocolate.