A mathematician at UNSW Sydney has introduced a groundbreaking method to solve one of algebra’s oldest unsolved problems: equations with variables raised to the fifth power or higher.
Honorary Professor Norman Wildberger, working with US computer scientist Dr. Dean Rubine, has developed a new approach that avoids using roots and irrational numbers.
Their work, published in The American Mathematical Monthly, offers a potential pathway to solving high-degree polynomial equations—a challenge that has eluded mathematicians for nearly 200 years. “Our solution reopens a previously closed book in mathematics history,” Wildberger said.
An ancient problem with modern implications
Polynomial equations, which include expressions like 1 + 4x – 3x² = 0, are essential in fields ranging from astronomy to computer science.
While formulas to solve lower-degree polynomials, such as quadratic, date back to ancient Babylon around 1800 BC, the general solution to fifth-degree equations, known as quintics, has remained out of reach since the 19th century.
I get why pure mathematicians like Prof. Norman Wildberger take issue with infinity, ∞. This assumption that 1/∞ IS zero and hence the interval (0,1/∞) seizes to tend to the POINT (0, 0) but IS actually the point (0, 0), is a fragile basis for pure mathematics.🫤 Problematic! pic.twitter.com/jztbCBhB2E
— Ovam Nomolos (@vmnmls) August 24, 2024
In 1832, French mathematician Évariste Galois proved that no universal formula exists for solving these higher-degree equations using traditional methods involving radicals.
Since then, most solutions have relied on numerical approximations. Wildberger argues those belong to applied math, not pure algebra.
Challenging the role of irrational numbers
At the center of Wildberger’s critique is the concept of irrational numbers, such as the square root of 2 or the cube root of 7. These values extend infinitely after the decimal point and never repeat, making them impossible to calculate exactly.
Visualisation of irrational nature of π. Its decimal representation is infinite and non-repeating, distinguishing it sharply from rational numbers, which either terminate or repeat in their decimal form. ✍️ pic.twitter.com/2bOzsSotIx
— Physics In History (@PhysInHistory) September 28, 2024
Wildberger explained that when we assume a number like the cube root of 7 “exists” in a formula, we are pretending that an endless decimal can be treated as a complete object. “You would need an infinite amount of work and a hard drive larger than the universe.”
This skepticism has shaped much of his career. Wildberger is known for rejecting traditional trigonometry and creating alternatives like rational trigonometry and universal hyperbolic geometry, which avoid irrational numbers altogether.
Power series provide a new path
His new method relies on power series—polynomials with many terms—instead of radicals. By truncating these series, the researchers produced accurate approximations to test their theory.
One test involved a 17th-century equation used by mathematician John Wallis to demonstrate Newton’s method. “Our solution worked beautifully,” Wildberger said.
The Geode may unlock discoveries
To support the approach, the team developed a new number array they call the “Geode,” an extension of the well-known Catalan numbers used in computer science, biology and game theory. These new sequences help represent geometric patterns and open the door to further mathematical exploration.
Catalan’s conjecture:
Catalan’s conjecture says that there is only one pair of powers of natural numbers that are consecutive, meaning that they differ by 1. This pair is 8 and 9, which are powers of 2 and 3 respectively.
In other words, there are no other solutions to the… pic.twitter.com/V79YIPw302
— Physics In History (@PhysInHistory) October 16, 2023
“We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years,” Wildberger said. “Really, there are so many other possibilities. This is only the start.”
The method also carries practical potential. Wildberger says it could lead to improved algorithms in computing, offering faster and more accurate ways to solve equations across many disciplines.
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