A Trillion Triangles
http://www.aimath.org/news/congruentnumbers/
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 of these more mysterious
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
a^2 so that a^2 -n and a^2 + 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 (therefore also the new computational result)
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 Millennium 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.
Contact Information
Media contact:
Estelle Basor
Deputy Director
American Institute of Mathematics
ebasor@aimath.org
(650) 845-2071
Research contact:
Bill Hart
Research Fellow
University of Warwick
W.B.Hart@warwick.ac.uk