News from the world of maths: 29 steps to universal (26/04/2006)

Think of a number, any number, and see if you can write it as the sum of square numbers: 13 = 2² + 3², 271 = 1² + 1² + 10² + 13², 4897582 = 6² + 95² + 2211²...

In 1770, Lagrange proved that every positive integer, no matter how large, can be written as a sum of at most four squares, x² + y² + z² + t². In the centuries since, mathematicians searched for other universal quadratic forms which could represent all the positive integers. Another 53 expressions, including 1x² + 2y² + 3z² + 5t², were found by Ramanujan in 1916. So many such universal forms exist, but how can you predict if a particular quadratic form is universal?

Now the brilliant young mathematician Manjul Bhargava and his colleague Jonathan Hanke have found a surprisingly simple result that completely solves the problem of finding and understanding universal quadratic forms. They found a shortcut to deciding if a quadratic form is universal — to check if the form represents every single positive integer, you only have to check it represents a mere 29 particular integers, the largest of which is 290. Bharghava and Hanke then went on to find every universal quadratic form (with four variables), all 6,436 of them. You can read about this surprising result in Ivars Peterson's excellent article in Science News.

Apart from Bhargava's brilliance as a mathematician (he was one of the youngest people to be made a full professor at just 28), he is also an accomplished musician. Both number theory and tabla playing may be viewed as the study of patterns, Bhargava told Peterson. "The goal of every number theorist and every tabla player," he explains in the article, "is to combine these patterns, carefully and creatively, so that they flow as a sequence of ideas, tell a story, and form a complete and beautiful piece."

If you're struggling with a sum of squares, try out this useful applet by Dario Alpern.