AI Tackles Erdős Problems * Science and Technology Times

New results from a collaboration between developer Neel Somani and OpenAI’s GPT‑5.2 Pro suggest that AI systems may now contribute original reasoning to long‑standing mathematical problems.
Several Erdős conjectures received new, formally verified solutions through this human‑AI workflow.
The work highlights a shift in how mathematicians may approach complex or obscure problems in the future.

AI Challenges Long‑Held Assumptions in Mathematics

For decades, the collection of more than a thousand Erdős problems has represented a demanding benchmark for mathematical creativity. These conjectures span number theory and combinatorics, often requiring subtle reasoning rather than brute‑force computation. Artificial intelligence was long assumed to be useful only for calculation or literature search, not for generating new proofs. Recent work by software engineer Neel Somani and the GPT‑5.2 Pro model suggests that this boundary may be shifting.

Somani set out to test the model’s reasoning abilities rather than its speed. He provided GPT‑5.2 Pro with Erdős Problem #397, a question involving integer solutions to a binomial identity. After roughly fifteen minutes, the model produced a complete argument proposing that one parameter could be chosen arbitrarily large, yielding infinitely many valid solutions. Somani verified the logic and formalized the proof using the Harmonic system before submitting it for expert review.

The result was accepted by mathematician Terence Tao, a rare endorsement in the field. The model’s reasoning differed from earlier partial attempts and included a concrete example demonstrating the identity. While the system did locate a related discussion from 2013, its final proof addressed the specific Erdős formulation more directly. This outcome marked one of the first instances in which an AI contributed a new, validated solution to an open mathematical problem.

A Second Breakthrough Strengthens the Case

Somani next turned to Erdős Problem #281, which concerns covering systems and the density of integers that fail to satisfy a sequence of congruences. The question asks whether this density can be made arbitrarily small under certain conditions. GPT‑5.2 Pro generated a new proof addressing the general case. When Somani shared the result, Tao described it as “perhaps the most unambiguous instance” of AI solving an open problem.

This distinction matters because earlier AI‑generated “solutions” often amounted to rediscovering known results. In this case, no prior published solution existed. The model produced a chain of reasoning that filled a genuine gap in the literature. Somani’s work suggests that AI may be particularly effective on problems that require persistence and careful combination of known theorems.

To better understand the model’s capabilities, Somani organized a dataset of GPT responses to 675 open Erdős problems. The majority—618—were summaries of existing literature. Seventeen were incorrect, twelve reproduced known results accurately, and three offered genuinely new solutions. Although the number is small, producing even a handful of publishable results is notable in a field where progress is often incremental.

Human Expertise Remains Central to the Workflow

Somani emphasizes that the AI did not work independently. His background in computer science, mathematics and quantitative research allowed him to craft precise prompts and evaluate the model’s output. The workflow involved identifying the problem, generating candidate proofs and formally verifying them using tools such as Harmonic. These systems help detect subtle logical errors that might otherwise go unnoticed.

Researchers like Tao suggest that AI may excel at the “long tail” of mathematical problems—those that are solvable but require extensive casework or unusual combinations of known results. Such problems often receive limited attention because they are time‑consuming rather than conceptually groundbreaking. AI systems can attempt thousands of these problems quickly, potentially clearing a backlog of unresolved questions. Human mathematicians can then focus on deeper structural challenges.

The broader implications extend beyond mathematics. If AI can reason through number‑theoretic problems, similar techniques could apply to fields such as cryptography, optimization or scientific modeling. Somani argues that the future of discovery will involve humans and AI working together, with formal verification ensuring reliability. This approach may reshape how researchers tackle complex problems across disciplines.