Mathematicians make a breakthrough on Gauss’s riddle, unsolved for 200 years

Mathematicians make a breakthrough on Gauss’s riddle, unsolved for 200 years

A solution to part of the Cohen-Lenstra conjecture helps resolve a long-standing mystery about quadratic forms

In his 1801 magnum opus Disquisitiones Arithmeticae, German mathematician Carl Friedrich Gauss wrote about a cyclical mystery. The puzzle involves quadratic forms , such as ax 2 + bxy + cy 2 . Setting the form equal to a number fixes it into an equation that can be plotted on an x - y graph—something many of us learned to do on graphing calculators in high school.

Gauss described a method to combine two of these forms to produce a third. He called the operation a “composition.” Using the method, he combined a quadratic form we’ll call Q with itself to find a new form, Q2. Composing Q2 with Q again, he got a third form, Q3. But as he kept repeating the steps over and over, he found that, after a finite number of iterations, the composition cycled through all possible forms and reset to the original form, Q.

Gauss could see that the cycle reset regardless of his starting form, but he couldn’t find any sort of rule governing the cycle’s length. Until recently, neither could any other mathematicians.

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In 1983 an idea inspired by Gauss’s discovery, the Cohen-Lenstra conjecture, claimed to be able to determine the average—not the exact—length of a cycle before it reset. That conjecture stood, generally accepted but unproven, for more than 40 years. Now Harvard University mathematician Aaron Landesman and Institute for Advanced Study Clay Research Fellow Ishan Levy have found a new framework that goes a long way toward proving it.

“This is a thing that many, many people work on,” says Melanie Wood, a mathematician at Harvard. It represents “a real breakthrough in our understanding of these types of questions.”