a young mathematician, has
solved a crucial chunk of a puzzle that has haunted number
theorists since the math legend Srinivasa Ramanujan scribbled his
revolutionary notions into a tattered notebook.
"In a nutshell, this [work] is the final chapter in one of the
most famous subjects in the story of Ramanujan," says Ken Ono, Mahlburg's graduate
advisor and an expert on Ramanujan's work. Ono is a Manasse
Professor of Letters and Science in mathematics.
"Mahlburg's achievement is a striking one, " agrees George
Andrews, a mathematics professor at Penn State University who has
also worked deeply with Ramanujan's ideas.
The father of modern number theory, Ramanujan died prematurely
in 1920 at the age of 32. The Indian mathematician's work is vast
but he is particularly famous for noticing curious patterns in the
way whole numbers can be broken down into sums of smaller numbers,
or "partitions." The number 4, for example, has five partitions
because it can be expressed in five ways, including 4, 3+1, 2+2,
1+1+2, and 1+1+1+1.
Ramanujan, who had little formal training in mathematics, made
partition lists for the first 200 integers and observed a peculiar
regularity. For any number that ends in 4 or 9, he found, the
number of partitions is always divisible by 5. Similarly, starting
at 5, the number of partitions for every seventh integer is a
multiple of 7, and, starting with 6, the partitions for every 11th
integer are a multiple of 11.
The finding was an intriguing one, says Richard Askey a emeritus
mathematics professor who also works with aspects of Ramanujan's
work. "There was no reason at all that multiplicative behaviors
should have anything to do with additive structures involved in
The strange numerical relationships Ramanujan discovered, now
called the three Ramanujan "congruences," mystified scores of
number theorists. During the Second World War, one mathematician
and physicist named Freeman Dyson began to search for more
elementary ways to prove Ramanujan's congruences. He developed a
tool, called a "rank," that allowed him to split partitions of
whole numbers into numerical groups of equal sizes. The idea worked
with 5 and 7 but did not extend to 11. Dyson postulated that there
must be a mathematical tool--what he jokingly called a
"crank"--that could apply to all three congruences.
Four decades later, Andrews and fellow mathematician Frank
Garvan discovered the elusive crank function and for the moment, at
least, the congruence chapter seemed complete.
But in a chance turn of events in the late nineties, Ono came
upon one of Ramanujan's original notebooks. Looking through the
illegible scrawl, he noticed an obscure numerical formula that
seemed to have no connection to partitions, but was strangely
associated with unrelated work Ono was doing at the time.
"I was floored," recalls Ono.
Following the lead, Ono quickly made the startling discovery
that partition congruences not only exist for the prime number 5, 7
and 11, but can be found for all larger primes. To prove this, Ono
found a connection between partition numbers and special
mathematical relationships called modular forms.
But now that Ono had unveiled infinite numbers of partition
congruences, the obvious question was whether the crank universally
applied to all of them. In what Ono calls "a fantastically clever
argument," Mahlburg has shown that it does.
A UW-Madison doctoral student, Mahlburg says he spent a year
manipulating "ugly, horribly complicated" numerical formulae, or
functions, that emerged when he applied the crank tool to various
prime numbers. "Though I was working with a large collection of
functions, under the surface I slowly began to see a uniformity
between them," says Mahlburg.
Building on Ono's work with modular forms, Mahlburg found that
instead of dividing numbers into equal groups, such as putting the
number 115 into five equal groups of 23 (which are not multiples of
5), the partition congruence idea still holds if numbers are broken
down differently. In other words, 115 could also break down as 25,
25, 25, 10 and 30. Since each part is a multiple of 5, it follows
that the sum of the parts is also a multiple of 5. Mahlburg shows
the idea extends to every prime number.
"This is an incredible result," says Askey.
Mahlburg's work completes the hunt for the crank function, says
Penn State's Andrews, but is only a "tidy beginning" to the quest
for simpler proofs of Ramanujan's findings. "Mahlburg has shown the
great depth of one particular well that Ramanujan drew interesting
things out of," Andrews adds, "but there are still plenty of wells
we don't understand."
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