Graduate Student Robert Bradshaw Helps Find Solutions to Thousand Year Old Mathematics Problem

Graduate student Robert Bradshaw is part of a group of mathematicians from North America, Europe, Australia, and South America who have resolved the first one trillion cases of a thousand year old mathematics problem.

The problem, first posed by the Persian mathematician al-Karaji (c.953 – c.1029), concerns the areas of right-angled triangles. 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” (see figure 1). The congruent numbers between one and twenty are 5, 6, 7, 13, 14, 15, and 20.

Figure 1: A 3-4-5 right triangle has area 1/2 x 3 x 4 = 6, so 6 is a congruent number.

The incredible task of finding the congruent numbers between one and one trillion was made possible by a clever technique for multiplying large numbers—numbers so enormous that if their digits were written out by hand they would stretch to the moon and back. As the numbers would not fit into the main memory of the available computers, the group made use of the computers’ hard drives. Each calculation was done twice on different computers using different algorithms by two independent teams. Bradshaw’s team used 128 gigabytes of accessible memory and 3 terabytes of storage space on UW’s Sage computers. They wrote their code at a workshop on L-functions in summer 2009.

One value of such problems is the new research developed by those looking for new ways to solve them, and the group’s efforts moved far beyond what others have done. As Bradshaw’s advisor William Stein told University Week, “Understanding this problem could be exactly what’s needed to understand many other interesting and important questions in mathematics.”

For more detail, see the American Institute of Mathematics press release.