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To Infinity and Beyond

Mathematicians hunt for patterns in the objects they study—"objects" such as the infinite set of numbers. But mathematicians must go beyond simply discovering new patterns, says Bruce Berndt, a mathematician at the University of Illinois. They must prove that the patterns they discern always hold true. 

 "It is not enough to observe that a pattern holds for, say, the first one million cases," Berndt says. "Mathematicians must show that the pattern holds for all cases, and usually this means an infinite number of cases. When a pattern is rigorously established to always hold, then this work can be applied by mathematicians and other scientists in their studies."

According to Berndt, Srinivasa Ramanujan (see accompanying article) was known for his work on "partitions"—an area of number theory that has yielded "many beautiful and useful patterns."

In mathematical language, the partition function p(n) is the number of ways you can write n as a sum of positive integers. For example, let us look at n = 4.  The number 4 can be written in five different ways as a sum of positive numbers or integers:

1+1+1+1

2+1+1

3+1

2+2

4

Thus, 4 has five different partitions, so we write p(4) = 5. Of course, as numbers get higher, the partition function p(n) quickly skyrockets, Berndt points out. For instance, 12 can be written in 77 different ways, which is to say, p(12) = 77. The number 200 has 3,972,999,029,388 partitions.

Ramanujan discovered some "beautiful theorems about the partition function," Berndt says.  In particular, he discovered three major patterns, which became known as Ramanujan's

"partition congruences."

As one example, Ramanujan found that beginning with the number 4, the partitions for every fifth number are divisible by 5. Thus, the number 4 has five partitions, the number 9 has 30 partitions, the number 14 has 135 partitions, and so on.

In other words, p(4), p(9), and p(14) are all divisible by 5.

According to Berndt, partition functions play a key role in many other fields, such as physics. For instance, theoretical physicists have used partitions in studying the distribution of particles in an area called statistical mechanics.

In a word, he says, Ramanujan's work on partitions was "epic."

Back to Raiders of the Lost Notebook.

By Doug Peterson
Fall/Winter 2006-07