See Inside April 2011

Cracking a Century-Old Enigma

Mathematicians unearth fractal counting patterns to explain a cryptic claim

Photo Researchers, Inc.

For someone who died at the age of 32, the largely self-taught Indian mathematician Srinivasa Ramanujan left behind an impressive legacy. Number theorists have now finally managed to make sense of one of his more enigmatic statements, written just one year before his death in 1920.

The statement concerned the deceptively simple concept of partitions. Partitions are subdivisions of a whole number into smaller ones. For example, for the number 5 there are seven options:

5 • 1 + 1 + 1 + 1 + 1 • 1 + 1 + 1 + 2
1 + 1 + 3 • 1 + 2 + 2 • 1 + 4 • 2 + 3

Mathematicians express this by saying p(5) = 7. For the number 6 there are 11 possibilities: p(6) = 11. As the number n increases, the partition number p(n) soon starts to grow very fast: for example, p(100) = 190,569,292, and p(1,000) is a 32-figure number.

For centuries mathematicians have struggled to make sense of partitions, in part by hunting for patterns that link them together. Ramanujan noticed that if you started with the number 9 and kept adding 5’s to that number, the partitions would all be divisible by 5. For example: p(9) = 30, p(9 + 5) = 135, p(9 + 10) = 490, and p(9 + 15) = 1,575. He posited that this pattern should go on forever and that similar patterns exist when 5 is replaced by 7 or 11, the next two prime numbers (primes are numbers that are divisible only by themselves or by 1), and also by powers of 5, 7 or 11. Thus, for instance, there should be an infinity of n’s at intervals of 53 such that all the corresponding p(n)’s should be divisible by 125. Then, in a nearly oracular tone, Ramanujan wrote that there should be no corresponding “simple properties” involving larger primes—in other words, there is no sequence of p(n)’s that are all divisible by 13, 17 or 19 and so on. In the years since, researchers have looked fruitlessly for patterns linking these higher primes.

In January, Ken Ono of Emory University and his collaborators finally found a solution: they described for the first time formulas linking n’s that come at intervals of the powers of 13 (13, 132, 133 ...) and of the higher primes. The formulas are not “simple,” in the sense that they do not say that the p(n)’s are divisible by powers of 13; instead they reveal relations among  the remainders of such divisions. For each prime, as the exponent grows, the formulas recur in ways that are reminiscent of fractals—structures in which patterns or shapes repeat identically at multiple different scales.

In a separate result also announced in January, Ono and another collaborator described the first formula that directly calculates p(n) for any n, a feat that had eluded number theorists for centuries.

Will the new discoveries have any practical use? Hard to predict, says George E. Andrews of Pennsylvania State University. “Deep understanding of underlying pure mathematics may take a while to filter into applications.” 

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