Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes.

Science News, June 18, 2005 by Erica Klarreich

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 1between 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 for a particular infinite collection of patterns. They concern partitions--ways of breaking up a number into a sum. The number 4, for instance, has five partitions (see box; p. 393). The number 5 has 7 partitions, and the number 6 has 11 partitions. The partition numbers quickly skyrocket: For instance, the partition number for 50 is 204,226 and for 200, it's 3,972,999,029,388.

To number theorists, partitions are among the most tantalizing objects in mathematics. However, even the simplest questions about the properties of partitions can be very hard to answer. For instance, no one has proved whether there are infinitely many partition numbers divisible by 3, although it's known that there are infinitely many partition numbers divisible by 2. There isn't much to distinguish a difficult question from one that can be easily solved, says Ken Ono, Mahlburg's adviser at Wisconsin.

While partitions were originally studied for their intrinsic interest, they have turned out to underlie a wide swath of mathematics, including some of the ideas that went into the proof of Fermat's last theorem by Andrew Wiles in 1993. Partitions also play a role in physics. For example, theoretical physicists employ them to explore the ways in which a collection of particles can be distributed among different energy configurations.

SURPRISING PATTERNS Fundamentally, partitions describe how to put together a number via addition. Yet, in 1919, Indian mathematician Srinivasa Ramanujan discovered that partitions have an unexpected connection to multiplication. They show patterns that rely on prime numbers, the building blocks for putting together a number via multiplication.

Ramanujan found that starting with the fourth partition number, which is 5, every fifth partition number is divisible by 5. For instance, the number 4 has 5 partitions, 9 has 30 partitions, and 14 has 135 partitions.

Ramanujan also discovered that starting with the 5th partition number, every 7th partition number is divisible by 7, and starting with the 6th partition number, every llth partition number is divisible by 11. These three patterns are called Ramanujan's partition congruences.

"These patterns are very unexpected," Ono says. "There's nothing about the definition of partitions that gives an easy explanation for why the three Ramanujan congruences exist" These congruences forge a link between two ways of expressing numbers--as sums and as products.

The numbers 5, 7, and 11 are consecutive primes, and the next prime is 13. So, extrapolating from Ramanujan's patterns, it makes sense to predict that, starting with the 7th partition number, every 13th partition number should be divisible by 13. Yet this is not so. After the three Ramanujan congruences, the pattern mysteriously breaks down.

For decades, mathematicians supposed that Ramanujan's three patterns were the only ones. In 1968, however, A. Oliver L. Atkin of the University of Illinois at Chicago discovered a few additional, much more complicated congruences. For instance, starting with the 237th partition number, every 17,303rd partition number is divisible by 13.

Then, in 2000, Ono astonished mathematicians by proving that partition congruences exist for every prime number (SN: 6/17/00, p. 396). This result was later generalized by Ono and Scott Ahlgren, now at the University of Illinois at Urbana-Champaign, to include all powers of primes. So, there are congruences not just for 5 but also for 52, 53, and so on.

Although Ramanujan proved that each member of a certain collection of partition numbers is divisible by 5, for example, his proof didn't give a way to break the number into five equal groups, or into groups of numbers all divisible by 5. In math, such a tangible breakdown is called a combinatorial proof. Ramanujan's work, and Ono's after it, relied on more-abstract proofs of divisibility.

Now, Mahlburg has come up with a combinatorial explanation for the unexpected divisibility patterns. His work completes a chain of ideas that was begun 6 decades ago by physicist Freeman Dyson of the Institute for Advanced Study in Princeton, N.J.

Mahlburg's work finally "gives a natural explanation for these congruences, which explains why they exist,' Ono says.