The sequence of partition numbers grows fast. But a general formula for calculating the number of partitions for any number n has been elusive.
Emory mathematician Ken Ono and colleagues will be announcing results today that include a finite, algebraic formula for partition numbers thanks to discovering that the sequence of partitions is fractal.
Partition numbers: In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition. In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
A EurekAlert press release appeared today, entitled: New math theories reveal the nature of numbers and people are already whispering “Fields Medal”. Obviously, like most press releases, this one is full of hyperbole and ridiculous sentences like, “the team was determined go beyond mere theories”, but the actual work being discussed is fascinating says one maths blogger.
The work of 18th-century mathematician Leonhard Euler led to the first recursive technique for computing the partition values of numbers. The method was slow, however, and impractical for large numbers. For the next 150 years, the method was only successfully implemented to compute the first 200 partition numbers. In the early 20th century, Srinivasa Ramanujan and G. H. Hardy invented the circle method, which yielded the first good approximation of the partitions for numbers beyond 200. They essentially gave up on trying to find an exact answer, and settled for an approximation.
Ramanujan also noted some strange patterns in partition numbers. In 1919 he wrote: “There appear to be corresponding properties in which the moduli are powers of 5, 7 or 11 … and no simple properties for any moduli involving primes other than these three.”
The legendary Indian mathematician died at the age of 32 before he could explain what he meant by this mysterious quote, now known as Ramanujan’s congruences.
In 1937, Hans Rademacher found an exact formula for calculating partition values. While the method was a big improvement over Euler’s exact formula, it required adding together infinitely many numbers that have infinitely many decimal places.
On Friday, Emory mathematician Ken Ono will unveil new theories that answer these famous old questions.
Ono and his research team have discovered that partition numbers behave like fractals. They have unlocked the divisibility properties of partitions, and developed a mathematical theory for “seeing” their infinitely repeatingsuperstructure. And they have devised the first finite formula to calculate the partitions of any number.
“Our work brings completely new ideas to the problems,” says Ono, who will explain the findings in a public lecture at 8 p.m. Friday on the Emory campus. “We prove that partition numbers are ‘fractal’ for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our ‘zooming’ procedure resolves several open conjectures, and it will change how mathematicians study partitions.” …….. “We found a function, that we call P, that is like a magical oracle,” Ono says. “I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It’s the finite, algebraic formula that we have all been looking for.”