A mathematician at the University of New South Wales (UNSW) has introduced a fresh approach to tackling one of algebra’s biggest challenges—solving higher-order polynomial equations, where the variable is raised to the power of five or more.
For centuries, mathematicians have been able to solve lower-degree polynomials, such as quadratics, cubics, and quartics. But in 1832, Évariste Galois showed that the usual methods fail for polynomials of degree five and above, and no general formula could be found.
Now, in 2025, nearly 200 years later, UNSW Honorary Professor Norman Wildberger believes he has cracked the problem with an entirely different approach—one that doesn’t rely on radicals (roots of numbers like square and cube roots). He argues that irrational numbers, which never end and never repeat, make calculations impossible to complete.
His method, developed alongside computer scientist Dr. Dean Rubine, uses power series—polynomials with an infinite number of terms—to approximate solutions, sidestepping the need for irrational numbers entirely.
At the heart of the discovery are Catalan numbers (𝐶𝑚)—a sequence that counts how many ways a polygon can be split into triangles. Mathematicians know that the series of Catalan numbers satisfies a quadratic equation.
Prof. Wildberger and Dr. Rubine expanded this idea to introduce hyper-Catalan numbers (𝐶𝐦), which count subdivisions of a polygon into different shapes, like triangles, quadrilaterals, and pentagons. Their research shows that the series of hyper-Catalan numbers also satisfies a polynomial equation with a distinct geometric pattern.
With this insight, they extended the method to solve general polynomial equations. By layering the series based on the number of faces in these shapes, they uncovered an extraordinary numerical pattern—the Geode.
“The Geode is a mysterious array that appears to underlie Catalan numerics,” says Prof. Wildberger.
This breakthrough isn’t just theoretical—it has real-world applications. Many scientific and computational problems rely on solving polynomial equations, and Prof. Wildberger’s method could lead to improved algorithms that avoid inefficient radical-based computations.
“This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas,” he explains.
His discovery has also opened new doors for mathematicians studying combinatorial sequences, sparking fresh questions about the structure and behavior of the Geode array.
“We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years,” Prof. Wildberger says.
With this new method, even quintic equations—polynomials of degree five—can now be tackled logically, and as such, the research continues to generate discussions in academic circles.
This article was generated with some help from AI and reviewed by an editor.
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