Artificial Intelligence Autonomously Solves a 60-Year-Old Mathematical Puzzle

2026-10-10 digital

Eindhoven, Saturday 10 October 2026
Eindhoven mathematician Tom Verhoeff used Anthropic’s Claude Opus 5.5 model to autonomously prove Lehmer’s 1965 permutation conjecture, shrinking a 150-page complex proof down to under 30 pages.

From Human Collaboration to Autonomous Discovery

The mathematical puzzle, first formulated in 1965 by American mathematician D.H. Lehmer, concerns permutations of identical items or dancers in a sequence where adjacent pairs swap positions [2][3][5]. While retired Eindhoven University of Technology (TU/e) mathematician and computer scientist Tom Verhoeff resolved the binary two-colour case in 2017 [1][2][3][5][6], the general case resisted all analytical attempts for six decades [2][3][5]. Initial efforts to leverage artificial intelligence models in March 2026 proved unsuccessful [1][2][3], following a period in late 2025 when top-tier models like OpenAI’s GPT-5.2 were still prone to elementary mathematical mistakes [3][6]. However, after working closely with earlier iterations such as Anthropic’s Claude Opus 4.7 in May 2026—which served as a diligent mathematical assistant—Verhoeff submitted an initial 150-page human-AI collaborative proof requiring over 100,000 lines of formal Lean verification code [1][2][3][4][5][6].

A Quantum Leap in Model Capabilities

The landscape shifted dramatically in September 2026 when Verhoeff tasked Anthropic’s newly released Claude Opus 5.5 model to tackle Lehmer’s conjecture from scratch [1][2][3][4][5][6]. Within a matter of hours, Opus 5.5 autonomously formulated two novel core concepts—blocking sequences and linking blocks—to construct an entirely independent, highly elegant proof [1][2][3][5][6]. This autonomous breakthrough delivered a text under 30 pages long, representing a page reduction of at least 80% compared to the original 150-page proof [1][2][3][4][5][6]. Furthermore, the formal verification footprint in the Lean proof assistant dropped from over 100,000 lines to approximately 35,000 lines, yielding an efficiency gain of approximately 65% in computer-checked logic code [1][2][3][4][5][6].

The Multi-Model Tech Stack Behind the Proof

The autonomous reasoning of Claude Opus 5.5 was supported by a broader computational workflow across specialized AI tools and hardware [1][2][3][5][6]. The DeepSeek V4.1 Flash model handled step-by-step conversions of reasoning into executable code, which was verified using concrete examples on TU/e’s Spike-1 supercomputer [1][2][3][5][6]. Formal Lean code generation was subsequently executed by Harmonic’s Aristotle system [1][2][3][5][6]. Highlighting the speed of execution, Verhoeff noted that the supercomputer’s raw processing power allowed researchers to cycle rapidly from AI output to human oversight and back into refined prompts [1][2][3][6]. On 9 October 2026, Verhoeff officially submitted the condensed proof as a preprint to arXiv (reference: https://arxiv.org/abs/2610.01240) for open peer evaluation [2][3][5][6].

Broader Ecosystem and Academic Shift

This landmark development coincides with a wider surge in automated scientific discovery, marked by OpenAI releasing hundreds of AI-generated mathematical proofs during the week of 5 October 2026 [3][5][6]. Reflecting on the rapid transition, Verhoeff emphasized that the advancement of AI models from basic assistants to autonomous grandmasters represents a paradigm shift far greater than the arrival of electronic calculators in the 1970s or computer algebra systems in the 1980s [1][2][3][5][6]. As AI transitions from a utility to an autonomous reasoning engine, deep tech ecosystems and academic institutions across the Benelux region face an urgent mandate to rethink foundational research workflows, software engineering, and university curricula [1][2][3][5][6].

Sources & Ecosystem Partners

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