September 22, 2009 - Mathematicians from North America, Europe,
Australia, and South America have resolved the first one trillion
cases of an ancient mathematics problem. The advance was made
possible by a clever technique for multiplying large numbers. The
numbers involved are so enormous that if their digits were written
out by hand they would stretch to the moon and back. The biggest
challenge was that these numbers could not even fit into the main
memory of the available computers, so the researchers had to make
extensive use of the computers' hard drives.
According to Brian Conrey, Director of the American Institute of
Mathematics, "Old problems like this may seem obscure, but they
generate a lot of interesting and useful research as people develop
new ways to attack them."
The problem, which was first posed more than a thousand years
ago, concerns the areas of right-angled triangles. The surprisingly
difficult problem is to determine which whole numbers can be the
area of a right-angled triangle whose sides are whole numbers or
fractions. The area of such a triangle is called a "congruent
number." For example, the 3-4-5 right triangle which students see
in geometry has area 1/2 x 3 x 4 = 6, so 6 is a congruent number.
The smallest congruent number is 5, which is the area of the right
triangle with sides 3/2, 20/3, and 41/6. The first few congruent
numbers are 5, 6, 7, 13, 14, 15, 20, and 21. Many congruent numbers
were known prior to the new calculation. For example, every number
in the sequence 5, 13, 21, 29, 37, ..., is a congruent number. But
other similar looking sequences, like 3, 11, 19, 27, 35, ...., are
more mysterious and each number has to be checked individually.
The calculation found 3,148,379,694 new congruent numbers up to
a trillion.
*Consequences, and future plans
Team member Bill Hart noted, "The difficult part was developing
a fast general library of computer code for doing these kinds of
calculations. Once we had that, it didn't take long to write the
specialized program needed for this particular computation." The
software used for the calculation is freely available, and anyone
with a larger computer can use it to break the team's record or do
other similar calculations.
In addition to the practical advances required for this result,
the answer also has theoretical implications. According to
mathematician Michael Rubinstein from the University of Waterloo,
"A few years ago we combined ideas from number theory and physics
to predict how congruent numbers behave statistically. I was very
pleased to see that our prediction was quite accurate." It was
Rubinstein who challenged the team to attempt this calculation.
Rubinstein's method predicts around 800 billion more congruent
numbers up to a quadrillion, a prediction that could be checked if
computers with a sufficiently large hard drive were available.
*History of the problem
The congruent number problem was first stated by the Persian
mathematician al-Karaji (c.953 - c.1029). His version did not
involve triangles, but instead was stated in terms of the square
numbers, the numbers that are squares of integers: 1, 4, 9, 16, 25,
36, 49, ..., or squares of rational numbers: 25/9, 49/100, 144/25,
etc. He asked: for which whole numbers n does there exist a square
a2 so that a2-n and a2+n are also squares? When this happens, n is
called a congruent number. The name comes from the fact that there
are three squares which are congruent modulo n. A major influence
on al-Karaji was the Arabic translations of the works of the Greek
mathematician Diophantus (c.210 - c.290) who posed similar
problems.
A small amount of progress was made in the next thousand years.
In 1225, Fibonacci (of "Fibonacci numbers" fame) showed that 5 and
7 were congruent numbers, and he stated, but did not prove, that 1
is not a congruent number. That proof was supplied by Fermat (of
"Fermat's last theorem" fame) in 1659. By 1915 the congruent
numbers less than 100 had been determined, and in 1952 Kurt Heegner
introduced deep mathematical techniques into the subject and proved
that all the prime numbers in the sequence 5, 13, 21, 29, ..., are
congruent. But by 1980 there were still cases smaller than 1000
that had not been resolved.
*Modern results
In 1982 Jerrold Tunnell of Rutgers University made significant
progress by exploiting the connection (first used by Heegner)
between congruent numbers and elliptic curves, mathematical objects
for which there is a well-established theory. He found a simple
formula for determining whether or not a number is a congruent
number. This allowed the first several thousand cases to be
resolved very quickly. One issue is that the complete validity of
his formula depends on the truth of a particular case of one of the
outstanding problems in mathematics known as the Birch and
Swinnerton-Dyer Conjecture. That conjecture is one of the seven
Millenium Prize Problems posed by the Clay Math Institute with a
prize of one million dollars.
*The computations
Results such as these are sometimes viewed with skepticism
because of the complexity of carrying out such a large calculation
and the potential for bugs in either the computer or the
programming. The researchers took particular care to verify their
results, doing the calculation twice, on different computers, using
different algorithms, written by two independent groups. The team
of Bill Hart (Warwick University, in England) and Gonzalo Tornaria
(Universidad de la Republica, in Uruguay) used the computer
"Selmer" at the University of Warwick. Selmer is funded by the
Engineering and Physical Sciences Research Council in the UK. Most
of their code was written during a workshop at the University of
Washington in June 2008.
The team of Mark Watkins (University of Sydney, in Australia),
David Harvey (Courant Institute, NYU, in New York) and Robert
Bradshaw (University of Washington, in Seattle) used the computer
"Sage" at the University of Washington. Sage is funded by the
National Science Foundation in the US. The team's code was
developed during a workshop at the Centro de Ciencias de Benasque
Pedro Pascual in Benasque, Spain, in July 2009. Both workshops were
supported by the American Institute of Mathematics through a
Focused Research Group grant from the National Science
Foundation.
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