What is Erdős Problem #1196?

Paul Erdős (1913–1996), a prolific mathematician, left behind a database of hundreds of unsolved conjectures. Problem #1196 is an “asymptotic” version of the Primitive Set Conjecture, originally posed in 1966 by Erdős together with fellow Hungarian mathematical legends András Sárközy and Endre Szemerédi.

A “primitive set” is a set of integers where no one number divides another. The primes are the canonical example. The conjecture asks about the behavior of a particular mathematical sum over primitive sets restricted to “large” numbers — essentially, how the sum behaves as you look further and further along the number line.

Lichtman had made the best previous partial progress on the problem, alongside collaborators Gorodetsky and Wong, but the full conjecture remained out of reach.

The Proof Nobody Thought To Write

What makes the GPT-5.4 result interesting is also how it solved it.

Since Erdős’ original 1935 paper establishing the foundations of this area, every mathematician who worked on primitive set problems — including Lichtman — had used the same conceptual “opening move”: transforming the problem from the discrete world of integers into the continuous world of real analysis. It was so natural an approach that it “obscured a technical possibility” that had been sitting in plain sight for 90 years, Lichtman said.

Instead of making the leap to analysis, GPT-5.4 stayed in the arithmetic realm and deployed the von Mangoldt function — a classical tool in number theory known primarily for its deep connections to the prime numbers and the Riemann zeta function — in an unexpected way. The function encodes a fundamental identity: the sum of von Mangoldt weights over divisors of any number n equals the logarithm of n, a statement equivalent to the unique factorization of integers into primes. That identity was the key needed to dissolve the analytic difficulties that had blocked every prior approach, the AI’s proof revealed.

“The closest analogy I would give would be that the main openings in chess were well-studied,” Lichtman said, “but AI discovers a new opening line that had been overlooked based on human aesthetics and convention.”

Is Mathematics Changing?

Over the last few years, AI has been used to prove increasingly abstract results, and that trend has not gone unnoticed inside the mathematical establishment. In 2026, at a major mathematical conference in Washington, D.C., there were reportedly plenty of nervous jokes about being made obsolete by AI, even if, on the record, people insisted that AI will be a helpmate to human mathematicians.

The U.S. Defense Advanced Research Projects Agency launched an initiative called expMath — Exponentiating Mathematics — to accelerate mathematical discovery using AI, structuring the program into two areas: advancing the state of the art in AI for mathematics, and evaluating the effectiveness of those AI systems. A startup called Math Inc. is reporting initial success in formalizing proofs, with its AI, Gauss, having formalized two complex proofs related to sphere-packing in higher dimensions by Fields Medalist Maryna Viazovska.

Meanwhile, the independent research community has demanded higher standards of evidence. In February 2026, 11 leading mathematicians published the “First Proof” initiative — ten mathematical questions that arose naturally during their own research, with answers encrypted and uploaded to a verification site, giving AI systems one week to attempt problems that had never appeared in any training dataset. The initiative was designed to short-circuit the data contamination problem that haunts AI math benchmarks. Preliminary results suggest current publicly available AI systems cannot yet reliably clear that bar autonomously — but that baseline may advance fast.

First Proof

The formal Lean verification of the Erdős #1196 proof is developing. If it’s successful, the result will join a growing list of AI-generated mathematics that has been certified at a competitive standard of rigor that mathematicians often apply to human proofs. That certification transforms an impressive-looking output into immutable mathematical data.

At the same time, there are also legitimate reasons for caution. AI systems currently score dramatically lower on open-ended research mathematics than on competition problems, and the “First Proof” challenge suggests genuine, consistent research-level autonomy is so far considerably elusive. The distinction between “finding an idea no human had considered” and “recombining patterns from training data in a novel way” is one that philosophers and mathematics could perhaps debate for years.

However, Jared Lichtman spent several years on a problem, and an AI spent 80 minutes on it, producing what he calls a proof from The Book. Whatever the philosophical fine print and consensus, that sentence may have sounded like science fiction 20 months ago. Today, it is simply Friday.