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Science Frontiers ONLINE No. 132: NOV-DEC 2000 |
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Puzzling Partitions
First, a quick primer on a fascinating mathematical byway called "partitions." A partition is a way in which a whole number can be expressed as the sum of positive integers. For example, 5 can be partitioned in seven ways:
5 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1+ 1 |
The number 4 has only five partitions. Check it out.
Historically, ordinary mortals saw no patterns in the number of partitions possessed by the parade of numbers until Ramanujan came along. He had in front of him a list of the number of partitions for each of the first 200 integers. They ranged from one (for 1) to 3,972,999,029,388 (for 200). [That of a computer is itself worthy of mention!]
Here is the order that Ramanujan perceived:
Starting with 5, the number of partitions for every seventh integer is a multiple of 7, and starting with 6, the number of partitions for every 11th integer is a multiple of 11. Moreover, similar relationships occur where the interval between the chosen integers is a power of 5, 7, or 11 or a product of these powers.
Ramanujan was able to prove that these curious patterns also hold for all higher numbers beyond 200.
Ramanujan's discovery came as quite a surprise to the world of mathematics, as did the strange roles of the three adjacent prime numbers 5, 7, and 11. Recently, though, K. Ono has gone beyond Ramanujan and proved that there are really an infinite number of relationships like the three found by Ramanujan.
(Peterson, Ivars; "The Power of Partitions," Science News, 157:396, 2000.)
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