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# Cracking a Century-Old Enigma

Mathematicians unearth fractal counting patterns to explain a cryptic claim

Srinivasa Ramanujan Image: 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.” 

This article was originally published with the title Cracking a Century-Old Enigma.

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1. 1. dbtinc 08:27 AM 4/14/11

Is it possible that this is of exactly no consequence thus expending truly useless calories? A brilliant mathematical mind wasted on things of absolutely no value. Sorry to sound so harsh but we live in a cruel world.

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2. 2. jtdwyer in reply to dbtinc 10:15 AM 4/14/11

Personally I agree, but meaningless incidental mathematical relations have sometimes been put to very good use in analytically describing events and processes that are otherwise not at all understood. Consider the standard model of particle physics, for example.

Personally, I hate math...

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3. 3. thevilleky 01:01 PM 4/14/11

I agree. We should stop doing anything with basic math. What's the point? We've discovered everything in the universe that can be discovered. I want my MTV.

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4. 4. Patristo 01:52 PM 4/14/11

http://www.archive.org/details/Ramanujan

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5. 5. cyborgv1 02:32 PM 4/14/11

This kind of math could be used for advanced encryption, compression, and data recovery which may someday be used by all.

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6. 6. NixKnacks 02:46 PM 4/14/11

Is it possible that reading your unenlightened and closed minded comment was of no consequence thus expending truly useless calories? A brilliant mathematical mind wasted, if only for a moment, on your mindless thoughts of no value. Sorry to sound so harsh, but we live in a cruel world.

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7. 7. osmosium 03:24 PM 4/14/11

dbtinc,

Thanks for an example of being useless. You seem to be an expert on that.

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8. 8. rluhlman 03:37 PM 4/14/11

Wow is this diatribe enlightening or what!!! Guess I'll go take a nap...

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9. 9. NixKnacks 03:46 PM 4/14/11

I just learned what diatribe means. That's pretty enlightening! :D

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10. 10. xristy 09:10 PM 4/14/11

Mr Castelvecchi and editors: In the future please include references to the published work that you're alluding to. Ms Harmon in her piece on Crayfish crawling includes a link the Science article on the research that she is reporting on. It should be standard practice in Scientific American to provide primary source references. It is a poor practice in popular science reporting to not include such references widens the gulf between active researchers and others who may be interested and capable of following the work beyond the level and details presented in the typical SA news summary.

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11. 11. bobgeezer in reply to dbtinc 09:59 PM 4/14/11

You have made the decision that his calories were wasted.

On what scientific grounds did you do that?

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12. 12. eco-steve 08:08 AM 4/16/11

Why should cosmology be ordered using maths formulae? The universe is essentially chaotic, and most processes are beyond full mathematical description, such as weather systems, which can only be approached statistically.
This might not please Pythagoricians, but the universe is incredibly complex.

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13. 13. jmblock2 01:16 AM 4/18/11

The universe is complex, but it isn't arbitrary. As far as we have observed, it fits a very rigid set of rules. Some of the rules we're still figuring out, but it really hasn't changed the ones we HAVE figured out. Newton's theory of gravity still works for low-speed bodies.

All of modern-day physics is based on abstract mathematics that fit our reality. Emily Noether found that any existing symmetry of an action corresponds to some conservation law. The separation of the electroweak force into the electromagnetic force and weak nuclear force occurs because of spontaneous breaking of on gauge group to another gauge group.

As far as this article goes, up until THIS year, we had no idea how such a simple concept of partitioning worked from a fundamental level. The formula may not be put into TI calculators anytime soon, but there is no doubt use to it in many fields. One possible use of this finding is insight into the possible configurations of molecules.

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14. 14. paulshaynes 11:04 AM 4/20/11

I thought usefulness was only ever sensibly measured a posteriori? Got to disagree with Hardy's Apology here (and most of the above) in that Applied Mathematics does come from Pure Mathematics sometimes! Gauss' modulo congruences for example have made the RSA guys rich. Good old Gauss.

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