A Centuries-Old "Impossible" Equation, Once Considered Unsolvable, Finally Has a Solution

The Debrief ^ | May 13, 2025 | Christopher Plain

[SunkenCiv]

A University of New South Wales (UNSW) Sydney mathematician has revealed the first successful solution of an 'impossible' equation once considered unsolvable.

Described as algebra's oldest problem, previous efforts to solve 'higher order' polynomial equations have consistently failed, leaving mathematicians without a critical tool. The new method solves that problem, potentially changing mathematics forever.

"Our solution reopens a previously closed book in mathematics history," said UNSW Honorary Professor Norman Wildberger, who led the research...

According to a statement from the University announcing the solution to algebra's oldest impossible equation, polynomials are represented by an established equation. For example, the degree two polynomial would be written as 1+ 4x – 3x2 = 0, where the variable "x" is raised to the second degree. However, the statement notes that whenever the variable is raised to five or higher, solving the equation has "historically proven elusive."

While solutions for two-degree polynomials have existed since the ancient Babylonians discovered them in 1800 BCE, the equation and its limits were not discovered until 1832 by French mathematician Évariste Galois. According to the release, Galois determined that the impossible equation was the limit and that "no general formula could solve them."

...Fortunately for the mathematics community... his work, including the new solution to an impossible equation, taps into special extensions of polynomials known as the "power series," which can have an infinite number of terms for the variable x without using radicals.

[TigersEye]

q = quarts
gal = gallons

It’s the new bra sizing measurement.

[Tymesup]

I remember there were formulas for 3rd and 4th degree polynomials in the CRC reference book. Complicated, I think the 4th degree formula led to a second formula.

Galois’ work helped Andrew Wiles prove Fermat’s Last Theorem. Since Fermat was before Galois, we still don’t know if Fermat had something or was mistaken.

[Tymesup]

Rationality doesn’t affect our ability to solve a polynomial. The first order equation x - 2^.5 = 0 is solved by x = 2^.5, an irrational number.

Using numerical methods - a power series - produces numerical answers that are as precise as we want them to be. However, that’s not a solution.

This article might have benefited from an example.

[SunkenCiv]

I pinged at least one person who is likely to know. :^)

[Mark17]

That guy had an advantage. He stayed at a Holiday Inn Express last night. 👍😜

[GingisK]

Surround the exponent with < sup> and < /sup and one gets x^y

[AdmSmith]

Norman is a nice person but he is against the https://en.wikipedia.org/wiki/Axiom_of_choice and the https://en.wikipedia.org/wiki/Axiom_of_infinity i.e. he has forgotten that they are axioms.

His history of math is entertaining (but skip the part about infinity and Babylonian math) https://www.youtube.com/playlist?list=PL34B589BE3014EAEB

[SunkenCiv]

Thanks A, but wait, skip the Babylonians?!? ;^) I did skip Ye’s opera “Nebuchadnezzar”...

[BarbM]

Oh, I see. And that little /\ sign is where they shove it up their arses.

thanks for the update.

[AdmSmith]

Wildberger has his own theory about Plimpton 322 a Babylonian clay tablet https://en.wikipedia.org/wiki/Plimpton_322

[SunkenCiv]

George probably faked it.

[SunkenCiv]

• The Babylonians Were Using Pythagoras' Theorem Over 1,000 Years Before He Was Born [08/07/2021]
• Incredible 3700-Year-Old Babylonian Clay Tablet Is World's Oldest Example of Applied Geometry [08/04/2021]
• Stonehenge builders used Pythagoras' theorem 2,000 years before Greek philosopher was born, experts [06/20/2018]
• Ancient Tablets Reveal Mathematical Achievements of Ancient Babylonian Culture [11/20/2010]
• Unprecedented mathematical knowledge found in (Minoan) Bronze Age wall paintings. [03/02/2006]