Numbers

My latest column for Quanta Magazine is about a clever technique for finding rational approximations to irrational numbers. The technique, developed by the German mathematician Gustav Dirichlet, works by covering the number line with tiny intervals centered at certain rational numbers.

But Dirichlet did better. He improved this method by figuring out how to shrink the intervals around their centers while still keeping the entire number line covered. As the intervals shrink, so does the distance to any irrational number we are trying to approximate. This means we’ll get better and better rational approximations, even using relatively small denominators. But we can’t shrink the intervals too quickly: Even though there are infinitely many of them, if the intervals get too small too fast they won’t cover the entire number line. In the battle between the infinitely large and the infinitely small, Dirichlet had to find the right balance to prevent some irrationals from slipping through the cracks.

Dirichlet’s technique explains why we can always find good rational approximations to irrational numbers using small denominators, like . Developed nearly 200 years ago, this technique ultimately led to the proposal of the Duffin-Schaeffer conjecture which was finally proved this past year.

You can read the full article here.

My latest column for Quanta Magazine is about the search for sums of cubes. While most integers are neither cubes nor the sum of two cubes, it is conjectured that most numbers can be written as the sum of three cubes. Finding those three cubes, however, can be a challenge.

For example, it was only this year we learned that the number 33 could be written as a sum of three cubes:

33 = 8,866,128,975,287,528³ + (−8,778,405,442,862,239)³ + (−2,736,111,468,807,040)³

What’s so hard about expressing numbers as a sum of three cubes?

It’s not hard to see that it combines the limited choices of the sum-of-squares problem with the infinite search space of the sum-of-integers problem. As with the squares, not every number is a cube. We can use numbers like 1 = 1³, 8 = 2³ and 125 = 5³, but we can never use 2, 3, 4, 10, 108 or most other numbers. But unlike squares, perfect cubes can be negative — for example, (-2)³ = -8, and (-4)³ = -64 — which means we can decrease our sum if we need to. This access to negative numbers gives us unlimited options for our sum, meaning that our search space, as in the sum-of-integers problem, is unbounded.

To learn more, read the full article, which is freely available here.