Ivars Peterson's MathTrek

March 24, 1997

Winning Partitions

In the world of mathematics, partitioning a whole number means breaking it up into a sum. For example, 5 can be partitioned in seven different ways: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1. Different arrangements of the parts don't count, so 2 + 1 + 2 is considered the same as 2 + 2 + 1.

The simple idea of partitions has developed into a significant branch of number theory. Indeed, "any time the number of ways of writing a number as the sum of other numbers arises, the theory of partitions can't be far off," says number theorist George E. Andrews of Pennsylvania State University.

Partitions display intriguing patterns. Notice that 5 has three partitions consisting entirely of odd numbers (5, 3 + 1, 1 + 1 + 1 + 1 + 1) and three partitions consisting of distinct numbers (5, 4 + 1, 3 + 2). One can prove that for a given whole number, the number of partitions in which all the parts are odd always equals the number of partitions in which all the parts are distinct. Leonhard Euler established this theorem in the eighteenth century.

One aspect of particular interest in the theory of partitions involves counting the number of different sums that generates a given positive integer. In general, the partition function p(n) is the number of partitions of n. Because there's only one way to get 1, p(1) = 1; because there are seven ways to get 5, p(5) = 7.

It turns out to be relatively straightforward to compute values of the partition function without having to write out all the sums. Here are the values of the partition function for the integers from 1 to 21: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, and 792. The values of p(n) grow rapidly as n gets bigger. The integer 200, for instance, has 3,972,999,029,388 partitions.

Mathematicians have expended considerable effort looking for patterns in the infinite sequence of numbers corresponding to the partition function. All sorts of surprising relationships arise.

One area of long-standing interest has been the parity of the partition function. In other words, when does the partition function take on odd or even values?

Recently, Ken Ono of the Institute for Advanced Study in Princeton proved some results involving the parity of the partition function and sets of integers known as arithmetic progressions. In an arithmetic progression, pairs of successive numbers have the same difference. For example, 1, 4, 7, 10, 13, 16 . . . is an arithmetic progression with a common difference of 3.

Ono proved that every arithmetic progression contains infinitely many integers for which the partition function of all those integers is even. The case for odd values, however, isn't quite settled. Ono could establish that every arithmetic progression contains infinitely many integers for which the partition functions of the integers is odd so long as there is at least one such integer in the sequence. Thus, there may be a loophole -- an arithmetic progression made up of integers that have only even-valued partition functions.

Nicholas K. Eriksson, a senior at Sentinel High School in Missoula, Mont., heard about the problem from Ono and began to study how to construct sets that guarantee the presence of odd values of the partition function. This effort formed the basis of his submission to the 56th Westinghouse Science Talent Search, and he emerged as the third-place winner when the awards were announced earlier this month.

Eriksson's work provided additional clues about when the partition function is odd. To furnish those clues, he delved into the same kinds of mathematics -- elliptic curves and modular forms -- that played a central role in the recent proof by Andrew Wiles of Fermat's Last Theorem.

Eriksson's paper describing his results has already been accepted for publication in a math journal.

Eriksson also happens to be a talented musician. He plays the alto saxophone, serves as head drum major in his school's marching band, and participates in the Missoula Youth Symphony. He was recently named outstanding soloist at the Lionel Hampton Jazz Festival.

It's a life of intricate harmony, both mathematical and musical.

References:

Andrews, George E. 1994. Number Theory. New York: Dover.

______. 1992. The reasonable and unreasonable effectiveness of number theory in statistical mechanics. Proceedings of Symposia in Applied Mathematics 46:21.

Eriksson, Nicholas K. 1999. q-series, elliptic curves, and odd values of the partition function. International Journal of Mathematics and Mathematical Sciences 22(March):55.

Kanigel, Robert. 1991. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Charles Scribner's Sons.

Mlot, C. 1997. Top 10 winners in top science contest. Science News 151(March 15):159.

Ono, Ken. 1995. Parity of the partition function. Electronic Research Announcements of the American Mathematical Society 1:35-42 (available at http://www.ams.org/era/).