Normally I would avoid maths problems where scientists start off by saying: "The numbers involved are so enormous that if their digits were written out by hand they would stretch to the moon and back."
Thankfully there are people who are challenged by such numbers and this week a group of such researchers said they, through a technique for multiplying large numbers, have figured out congruent numbers up to a trillion. Apparently no one had taken them beyond a billion for some reason.
In case you were wondering, a the first few congruent numbers are 5, 6, 7, 13, 14, 15, 20, and 21.
Many congruent numbers were known prior to the new research. 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, the reseachers said in a statement."
According to the researchers the biggest challenge in counting to the trillion mark 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.
"The computers the group used included the Warwick University's Selmer, a 4x Quad core AMD K10 operating at 2.4 GHz with 128GB Ram and a 1.5TB Hard Drive. The other was the National Science Foundation's Sage, a 4 x 6 core Intel Xeon CPU X7460 running at 2.66GHz with 128 GB ECC Ram and 2.7 TB NFS mounted hard drive storage located on the University of Washington site."
The problem, which was first posed more than a thousand years ago, concerns the areas of right-angled triangles. The difficult part 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, researchers stated.
"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," said Brian Conrey, Director of the American Institute of Mathematics in a release.
If you want to read the white paper that goes into excruciating detail about the exact calculations, click here.
The results, the researchers from the University of Sydney, in Australia, Courant Institute, NYU, in New York and the University of Washington, in Seattle, say will help answer some seriously geeky mathmatical arguments about congruent numbers It also opens the door to calculating higher numbers if others have bigger computers to handle it. If you are in to such things, it's pretty amazing.