Australia: Mathematicians Solve a Reputedly Insoluble Equation and Revolutionize Their Field

A lifelong dream has come true. While researchers around the globe have been puzzled for two centuries over how to solve higher-degree polynomial equations, a mathematician from the University of New South Wales in Australia has found the answer. Norman Wildberger seems to have struck upon a method that has never been tried in this context... and it has yielded results.

Polynomial equations are algebraic equations with one variable, x. If this reminds you of middle or high school math classes, that’s only part of the explanation. In these equations, x is associated with non-negative exponents, such as x². While scientists know how to solve such polynomial equations, they have previously been unable to do so beyond a certain level of complexity, especially when there are more than five different exponents on the variable. Norman Wildberger has addressed this issue.

Catalan Numbers to the Rescue of a 200-Year-Old Riddle

In a study published in The American Mathematical Monthly in June 2024, mathematician Norman Wildberger and his computer science colleague Dean Rubine detail their findings. The two men decided to start from the approach of Catalan numbers, generally used to determine how many triangles can be cut from polygons and in what ways.

This approach is unconventional. For simpler polynomial equations, mathematicians agree to use radical expressions—like square or cube roots—to solve them. This has never worked for higher-degree polynomial equations. “It is commonly accepted that Catalan numbers are closely connected to the quadratic equation [degree 2 polynomial],” explains Norman Wildberger. “Our innovation is to say that to solve complex equations, we need to find complex equivalents to the Catalan numbers.”

An Even More Unexpected Discovery with Potentially Unprecedented Consequences

This idea proved fruitful. The two researchers succeeded in solving higher-degree polynomial equations. But they did not stop there. They believe they have discovered a new mathematical structure they named “the Geode,” which seems to be a foundational base for Catalan numbers.

These new discoveries and the emergence of new concepts could completely revolutionize the world of mathematics. But it doesn’t stop there! Biology could utilize these new theories, particularly in the study of RNA.