Mathematicians from North America, Europe, Australia, and South America have resolved the first one trillion cases of an ancient mathematics problem on congruent numbers. The advance, which included work by David Harvey, an assistant professor at New York University’s Courant Institute of Mathematical Sciences, was achieved through a complex technique for multiplying large numbers.
The problem, first posed more than 1000 years ago, concerns the areas of right-angled triangles. A congruent number is a whole number equal to the area of a right triangle. The surprisingly difficult problem is to determine which whole numbers can be the area of a right-angled triangle whose sides are either whole numbers or fractions. For example, the 3-4-5 right triangle 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 this 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 new calculation found 3,148,379,694 new congruent numbers up to a trillion. The quantity of numbers involved in this calculation is significant — if their digits were written out by hand, they would stretch to the moon and back.
The congruent number problem was first stated by the Persian mathematician al-Karaji in the 10th century. His version did not involve triangles, but instead was stated in terms of the square numbers. In the 13th century, Italian mathematician Fibonacci showed that 5 and 7 were congruent numbers, and he stated, but didn’t prove, that 1 is not a congruent number. That proof was supplied by France’s Pierre de Fermat in 1659. By 1915, the congruent numbers less than 100 had been determined, but by 1980 there were still cases smaller than 1000 that had not been resolved. In 1982, Rutgers University mathematician Jerrold Tunnell 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.
The research team also included mathematicians from Warwick University (England), Universidad de la Republica (Uruguay), the University of Sydney (Australia), and the University of Washington in Seattle. The work was supported by the American Institute of Mathematics through a Focused Research Group grant from the National Science Foundation.