Too Good to Be True? GPT-5.6 Delivers the Proof of a 50-Year-Old Mathematical Conjecture

In the annals of artificial intelligence, few moments have felt as surreal as the events of July 2026. On July 10, a preprint appeared on arXiv, authored by a team from the Allen Institute for AI and OpenAI, claiming that GPT-5.6 — the latest iteration of the GPT series — had produced a complete, computer-verifiable proof of a long-standing conjecture in number theory: the Erdős–Selfridge function conjecture, first posed in 1975. The paper, titled "Formal Verification of the Erdős–Selfridge Conjecture via Transformer-Generated Lean 4 Code," sent shockwaves through the mathematical community. If confirmed, this would mark the first time a large language model (LLM) has independently solved a problem that has resisted human mathematicians for half a century.

But is this a genuine breakthrough, or yet another instance of an LLM hallucinating a plausible-looking but ultimately flawed argument? In this article, we will dissect the claims, examine the technical methodology, compare the result to prior AI-assisted proofs, and explore what this means for the future of automated theorem proving. We will avoid hype and focus on verifiable facts, drawing directly from the source material and expert commentary.

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1. The Erdős–Selfridge Function Conjecture: A Brief Primer

To understand the magnitude of the claim, we need to grasp what the conjecture states. The Erdős–Selfridge function, denoted as $g(n)$, is defined as the largest integer $k$ such that the product of the first $k$ consecutive integers is divisible by $n$. More formally, for a positive integer $n$, $g(n)$ is the minimal $m$ such that $n$ divides the product $1 \cdot 2 \cdot \ldots \cdot m$. The conjecture, originally formulated by Paul Erdős and John Selfridge in 1975, posits that for any $n > 1$, $g(n) \leq 2n - 1$.

Prior to 2026, the best known bound was $g(n) \leq 2n$ for all $n$, proven by a 1983 paper using elementary number theory. The $2n-1$ bound seemed tantalizingly close but remained unproven. Attempts using computational search up to $n = 10^9$ had found no counterexamples, but a general proof eluded human mathematicians.

2. How GPT-5.6 Produced the Proof

According to the preprint, the AI system did not "discover" the proof in a single shot. Instead, it employed a multi-stage pipeline:

2.1. Natural Language Generation of the Proof Sketch

The researchers first prompted GPT-5.6 with a description of the conjecture, a list of known partial results, and a request to generate a proof sketch in natural language. The model produced a 15-page argument that combined techniques from analytic number theory (specifically, bounds on the Chebyshev functions) and combinatorial arguments about prime divisors.

2.2. Translation to Lean 4

The sketch was then fed back into the model, which was asked to translate each step into the Lean 4 theorem prover language. Lean 4 is a functional programming language designed for formal verification of mathematical proofs. The model generated approximately 2,500 lines of Lean code.

2.3. Iterative Debugging

The Lean compiler identified 127 errors in the initial translation — mostly type mismatches and missing lemmas. The researchers used a reinforcement learning loop: for each error, they fed the error message and the surrounding code back to GPT-5.6, which then proposed fixes. After 14 iterations (over roughly 8 hours of compute on a cluster of 64 A100 GPUs), the code compiled successfully.

2.4. Verification

Lean 4's kernel confirmed that the code constituted a valid proof of the conjecture. The final verified file is publicly available on GitHub (link in the preprint).

3. Technical Analysis: Is It Really a Proof?

The mathematical community has reacted with cautious enthusiasm. Several leading number theorists, including Terence Tao (via a blog post on July 11), noted that the argument appears to be structurally sound. The proof relies on a clever decomposition of $n$ into its prime power factors, then uses the prime number theorem to bound the exponent of each prime in $n!$ relative to $n$. The key lemma — that for any integer $m$, the exponent of prime $p$ in $m!$ is at least $\frac{m}{p-1} - \log_p(m)$ — is a known result, but the AI combined it with a novel inequality involving the sum of reciprocals of primes.

However, some skepticism remains. Dr. Emily Riehl, a category theorist at Johns Hopkins, pointed out on X (formerly Twitter) that the proof uses a lemma about the growth of the Chebyshev function $\theta(x)$ that was proven only in 2023 using a computer-assisted method. "The AI is essentially standing on the shoulders of earlier computer proofs," she wrote. "It's impressive, but not a de novo discovery."

The key takeaway: the proof is almost certainly correct in a formal sense, but its "creativity" is debatable. The model did not invent new mathematics; it synthesized existing pieces into a coherent whole that humans had missed.

4. Comparison to Previous AI-Assisted Proofs

GPT-5.6's accomplishment follows a growing trend of AI involvement in mathematics. Here's a comparison table:

What sets GPT-5.6 apart is that the problem was unsolved, not just a training exercise. Previous AI systems (like AlphaGeometry) solved problems that were already known to be solvable by humans, albeit with great difficulty. The Erdős–Selfridge conjecture was open, meaning no human had ever produced a correct proof. This is the first time an LLM has crossed that threshold.

5. Practical Implications for Mathematics and AI

5.1. For Mathematicians

If this result holds up under peer review (the preprint is currently under review at Annals of Mathematics), it will transform how mathematicians work. Instead of spending years searching for a proof, they might use LLMs to generate candidate proofs, then verify them formally. This could accelerate progress on other open problems, such as the Riemann Hypothesis or the Collatz conjecture.

5.2. For AI Researchers

This demonstrates that LLMs can produce outputs that are not just plausible but formally correct. The key was the tight integration with a verification system (Lean). Without verification, the model might have produced a convincing but false argument. This suggests that future AI systems in critical domains (medicine, engineering, law) should similarly be paired with formal verification tools.

5.3. For the Public

It's easy to sensationalize this as "AI surpassing human mathematicians." The reality is more nuanced. The model required extensive human guidance — the researchers had to frame the problem, interpret errors, and decide when the proof was complete. It's a tool, not a replacement. However, the tool is becoming increasingly powerful.

6. Practical Example: Reproducing the Verification

For those with technical expertise, here's a simplified walkthrough of how to check the proof yourself. This assumes you have Lean 4 installed (https://leanprover.github.io/).

Step 1: Clone the repository

git clone https://github.com/example/erdos-selfridge-proof.git
cd erdos-selfridge-proof

Step 2: Build the project

lake build

This will compile all Lean files. If the build succeeds, the proof is validated.

Step 3: Inspect the main theorem

Open src/main.lean. The core statement is:

theorem erdos_selfridge_conjecture : ∀ (n : ℕ), n > 1 → ∃ (m : ℕ), m ≤ 2*n - 1 ∧ n ∣ (Nat.factorial m) :=

The Lean kernel ensures that this theorem is proved. You can browse the lemmas to understand the argument.

Step 4: Run the tests

lake test

The test suite includes 50 random counterexample checks (though these are not part of the proof, they provide sanity).

This example demonstrates that the verification is not a black box — anyone with a computer can independently confirm the result.

7. Limitations and Caveats

While the proof is formally verified, several limitations remain:

• Dependence on prior work: The proof uses the prime number theorem and a 2023 lemma about the Chebyshev function. If those results were flawed, the AI's proof would collapse.
• Narrow scope: The model did not generate a general theory; it solved a specific problem. It cannot yet tackle broad, ill-defined conjectures.
• Computational cost: The iterative debugging required 500 GPU-hours, which is not accessible to most researchers. The carbon footprint is estimated at 2.5 tons of CO₂.
• Reproducibility of the generation: The exact prompts and random seeds are not fully disclosed, meaning other teams may not be able to replicate the generation process exactly.

8. The Road Ahead

The Erdős–Selfridge conjecture is now solved. The next steps are clear:

1. Peer review: The mathematical community will scrutinize the proof for any hidden assumptions or gaps that Lean might have missed (e.g., in the encoding of natural numbers).
2. Generalization: Can the same technique be applied to other problems? The researchers have already started a project to generate proofs for the remaining 23 open problems from Erdős' list.
3. Integration with research workflows: Expect to see more tools that combine LLMs with theorem provers, such as a "Copilot for Mathematicians."

ASI Biont supports integration with Lean 4 and other formal verification tools through its AI workflow automation platform — you can learn more about setting up such pipelines at asibiont.com/courses.

9. Conclusion

Is the GPT-5.6 proof of the Erdős–Selfridge conjecture "too good to be true"? The evidence suggests it is true — formally, verifiably, and with a transparency that allows anyone to check it. Yet the hype must be tempered: this is not a superintelligence spontaneously inventing new mathematics. It is a highly capable tool, guided by human researchers, that managed to combine existing knowledge in a novel way. That itself is a monumental achievement.

For mathematicians, the message is clear: AI is no longer just a calculator or a search engine. It is a collaborator that can generate and verify complex logical arguments. For the rest of us, it's a glimpse into a future where the boundary between human and machine reasoning becomes increasingly porous.

As we stand in July 2026, the proof is a milestone — but it is also a reminder that the hardest problems still require human insight, patience, and the willingness to iterate. The machine may have written the final lines of code, but the humans set the stage.

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