Mathematicians make a breakthrough on Gauss’s riddle, unsolved for 200 years | Scientific American
August 5, 2026<br>6 min read<br>Add Us On GoogleAdd SciAm<br>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
By Lyndie Chiou edited by Clara Moskowitz
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In his 1801 magnum opus Disquisitiones Arithmeticae, German mathematician Carl Friedrich Gauss wrote about a cyclical mystery. The puzzle involves quadratic forms, such as ax2 + bxy + cy2. 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.<br>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.<br>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.<br>On supporting science journalism<br>If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<br>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.<br>“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.”<br>Also remarkable is that although the question focuses on a single topic—quadratic forms—the proof flows among different branches of math, drawing insights from across the mathematical kingdom, including statistics, geometry, algebra and a topic called homotopy theory, which studies what stays the same about a shape as you stretch and squish it like you would dough.<br>From Deterministic to Probabilistic<br>After more than 150 years of little progress, the first foothold in Gauss’s problem came in 1983, when French mathematician Henri Cohen and Dutch mathematician Hendrik Lenstra surprised their colleagues by conjecturing that the cycle reset could be explained using probability. The idea wasn’t entirely new; randomness is the force behind many profound ideas in number theory, including the twin prime conjecture and the Riemann hypothesis.<br>A computational number theorist, Cohen calculated enormous tables of quadratic compositions and was the first to spot a pattern hidden within. Lenstra supplied the theory explaining the existence of that pattern. They didn’t directly answer Gauss’s question but instead a narrower one involving prime numbers, which are only divisible by themselves and 1.<br>Pick a prime number, p, greater than 2, they wrote. Across all the families of related quadratic forms, how many take exactly p steps to reset? Cohen and Lenstra’s answer: on average, only one family—no matter which prime you picked. Their conjecture became one of three major conjectures that have since defined arithmetic statistics—the study of how collections of mathematical objects behave on average, even though no formula predicts how a single object behaves.<br>The next step came in 2009, when mathematicians Jordan Ellenberg, Akshay Venkatesh and Craig Westerland posted a preprint paper that claimed to prove a weaker version of the conjecture for every prime. Their work took place in the world of function fields—the geometric cousins to Gauss’s number-based approach. The paper was a tour de force, composed of sections that concerned seemingly unrelated areas of math. Landesman read the work when he was a graduate student at Stanford University. “It was the most...