A Mathematician and Two AI Models Settle a Question John Conway Left Open
Joe Shipman, a mathematician known for his work on constructibility and the foundations of mathematics, has posted a proof that the general quintic equation can be solved using a marked ruler and compass. The result, shared on the FOM (Foundations of Mathematics) mailing list this week, resolves a problem he had discussed for years with the late John Conway, who believed such a construction existed.
The general quintic, a polynomial equation of degree five, has a famous distinction in mathematics. The Abel-Ruffini theorem proves it cannot be solved using only radicals, the root extractions that work for equations up to degree four. But radicals are not the only tools available to a geometer. A marked ruler, used in what is called a neusis construction, extends the set of things that can be built with compass and straightedge. Shipman's proof shows that extension is sufficient to reach the quintic.
The Construction, Stated Simply
The method has three components. First, a Tschirnhaus transformation removes the x-squared and x-fourth terms from the general quintic, reducing it to a simpler form. This is a standard technique in algebra, but applying it in the context of geometric construction requires care.
Second, and this is the core of the proof, a double neusis construction solves the reduced equation. A neusis involves aligning a marked ruler between two curves, with the constraint that the distance between the marks equals a fixed unit. Shipman uses a version where the compass acts as a divider, maintaining the same unit radius as the marks on the ruler. One such construction handles a cubic factor; a second neusis handles the remaining quintic structure.
Third, a sequence of square root extractions completes the solution. These are constructible by classical compass and straightedge methods, so no additional tools beyond the marked ruler are needed.
Conway, who passed away in 2020, had discussed this problem with Shipman repeatedly. "He'd have been so pleased to see I finally found the construction he was sure was there," Shipman wrote in his post. The proof settles a question that had been open in the constructibility literature for decades.
Why This Was Hard
The Abel-Ruffini theorem tells you that radicals alone won't work. But it doesn't tell you what additional tools will work, or how to find the right construction. The difficulty, as Shipman describes it, was not writing down the algebra but understanding why earlier searches for the construction had failed.
Shipman used algebraic geometry to analyze the structure of the problem. He needed to understand which types of constructions could potentially reach the quintic, and which types were provably insufficient. That required computing Galois groups, factoring polynomials, and extracting arithmetical information from thousands of test equations. For an algebraic geometer, those computations alone would be valuable. But Shipman needed to go further and use the results to guide his search toward the right class of constructions.
The algebraic geometry was the part that took time. Not the manipulation of equations, but the structural understanding of why certain approaches were doomed and where the solution was likely to hide.
AI as a Research Accelerant
Shipman credits Claude Opus and ChatGPT Sol with helping him through the algebraic and computational parts of the work. He is precise about what the tools did and what they did not do. The LLMs did not find the proof. They helped with the algebra that Shipman needed to perform in order to find the proof himself.
"If I'd been a tenured professor I maybe could have done it in a year of work without them, but I never had that year," he wrote. The tools compressed what would have been a lengthy computation-heavy phase into something manageable. He estimates the speedup on algorithm development at roughly 10x.
The practical impact was enabling Shipman to run large numbers of experiments quickly. Computing Galois groups and factorizations for thousands of equations would have been enough on its own. But Shipman needed to learn the algebraic geometry as well, and he found the LLMs well-suited to that kind of structured technical learning. They could explain concepts, generate examples, and help him reason about the geometry in ways that textbooks alone would have made slower.
This is a different use of AI in mathematics than the automated theorem proving that has grabbed headlines. Shipman did not ask an LLM to produce a proof. He used LLMs to accelerate the human process of exploration, computation, and structural understanding that led him to the proof. The distinction matters because it points to a realistic near-term role for AI in mathematical research: not replacing the mathematician, but giving them更快 access to the computations and explanations they need to think clearly about a hard problem.
What This Means for Constructibility
The result extends the classical theory of geometric constructibility. The original theory, developed in the 19th century, characterized which numbers and which polynomial roots could be constructed with compass and straightedge. Adding the marked ruler expands the reachable set. Shipman's proof places the general quintic within that expanded set.
For algebraists, the result is a clean resolution of a long-standing open question. For geometers, it demonstrates that neusis constructions are more powerful than some had assumed. And for the broader mathematical community, it offers a concrete example of how AI tools can fit into serious research workflows without displacing human judgment.
Shipman's post is available on the FOM mailing list. The proof itself has not yet been peer-reviewed in a journal, but the construction is explicit enough that other mathematicians can verify the algebra independently. The question is no longer whether the general quintic is solvable with a marked ruler and compass. It is solved. What remains is the formal work of publication and the broader conversation about what this means for the boundaries of constructibility.