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Pieces of Numbers

A proof brings closure to a dramatic tale of partitions and primes

10:54am, June 14, 2005

In the realm of mathematics, it's hard to imagine anything more basic than the counting numbers: 1, 2, 3, and so on. Yet this set of mathematical objects abounds with beautiful and unexpected patterns. For example, pick any number and double it. You'll always find a prime number—a number divisible only by itself and by 1—between that number and its double. As another case in point, primes that leave a remainder of 1 when divided by 4 can always be expressed as the sum of two squares. Now, a mathematics graduate student has put what may be the final piece into the picture of one of the most surprising patterns of all.

Working despite his adviser's warnings that the problem was exceedingly difficult, Karl Mahlburg of the University of Wisconsin–Madison has come up with an explanation f

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