A Trillion Triangles
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
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 × 3 × 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 3-4-5 triangle has area 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
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
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.
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.
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
so that a2-n
are also squares? When this happens, n is called a congruent
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.
Al-Fakhri fi'l-jabr wa'l-muqabala,
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.
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
Research Council in the UK.
Most of their code was written
during a workshop at the University of Washington in
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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