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

In 1801, German mathematician Carl Friedrich Gauss penned his magnum opus, Disquisitiones Arithmeticae, in which he introduced a fascinating cyclical mystery involving quadratic forms. Quadratic forms are expressions such as ax^2 + bxy + cy^2, and when set equal to a number, they can be plotted on an x-y graph. Gauss devised a method, termed "composition," to combine two quadratic forms and generate a third.

Repeating this process, he observed that after a finite number of iterations, the forms cycled through all possible combinations and returned to the original form. However, he couldn't pinpoint a rule governing the cycle's length. This long-standing enigma remained unsolved for over 200 years until recently.

In 1983, French mathematician Henri Cohen and Dutch mathematician Hendrik Lenstra introduced the Cohen-Lenstra conjecture, claiming that the average length of this cycle could be determined using probability. Although the conjecture stood unproven for more than four decades, Harvard University mathematician Aaron Landesman and Clay Research Fellow Ishan Levy recently discovered a new framework that brings us closer to proving it.

This breakthrough is particularly remarkable because, despite focusing on a single topic—quadratic forms—the proof spans various branches of mathematics, including statistics, geometry, algebra, and homotopy theory. Homotopy theory investigates the invariant properties of shapes when they undergo stretching and squishing.

The journey to this solution began in 1983 when Cohen and Lenstra conjectured that the cycle reset could be explained using probability. Their idea, though not entirely new, gained traction due to the role of randomness in profound ideas within number theory. Cohen used computational methods to identify patterns within quadratic compositions, while Lenstra supplied the theoretical foundation for these patterns.

In 2009, mathematicians Jordan Ellenberg, Akshay Venkatesh, and Craig Westerland published a preprint paper that claimed to prove a weaker version of the conjecture for every prime number. Their work incorporated insights from diverse mathematical fields, such as function fields, algebraic geometry, topology, combinatorics, homological algebra, and probability.

Ellenberg, one of the paper's authors, found the interdisciplinary approach inspiring, comparing it to a Reese's Peanut Butter Cup, with each section representing a distinct area of mathematics.

The team sensed they were on the verge of a significant breakthrough. The group completion theorem from homotopy theory seemed to offer the key to unraveling the problem. This theorem, an essential result in homotopy theory, predicts that as a collection of "spaces" (sets of points or braids) grows, it will eventually settle into a fixed, repeating pattern.

Intrigued by this analogy, Ellenberg, Venkatesh, and Westerland developed a new proof in 2012, but it was later found to contain a subtle flaw. The authors subsequently retracted their paper, acknowledging the discrepancy. Despite this setback, the researchers' work represents a significant stride toward resolving Gauss's riddle.