Five Mathematicians Capture Record Number of Congruent Numbers

October 1, 2009

Five mathematicians have captured a record number of congruent numbers, resolving the first trillion cases of an ancient mathematics problem concerning the areas of right-angled triangles.

The surprisingly difficult problem involves determining 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 well-known 3-4-5 right triangle has integer sides and area 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 team used advanced computer techniques for multiplying and storing enormous numbers to accomplish the feat.

"The difficult part was developing a fast general library of computer code for doing these kinds of calculations," observed Bill Hart (University of Warwick). "Once we had that, it didn't take long to write the specialized program."

Hart and Gonzalo Tornaria (Universidad de la Republica, in Uruguay) used the computer Selmer at the University of Warwick. Mark Watkins (University of Sydney), David Harvey (New York University), and Robert Bradshaw (University of Washington, in Seattle) used the computer Sage at the University of Washington.

"Old problems like this may seem obscure," noted Brian Conrey, director of the American Institute of Mathematics, "but they generate a lot of interesting and useful research as people develop new ways to attack them."

"A few years ago we combined ideas from number theory and physics to predict how congruent numbers behave statistically," mathematician Michael Rubinstein (University of Waterloo) said. "I was very pleased to see that our prediction was quite accurate."

The mathematicians' paper is titled "Congruent Number Theta Coefficients to 10^12."

Source: American Institute of Mathematics, Sept. 22, 2009; ScienceNow, Sept. 23, 2009.