A Trillion Triangles

Mathematicians have resolved the first one trillion cases of an ancient mathematics problem.

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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 × 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 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.

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.