Solving 20 Erdős Problems with 20 Codex Accounts Running in Parallel
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An experimental approach to solving complex mathematical problems attributed to Paul Erdős by utilizing 20 OpenAI Codex accounts running in parallel to increase computational breadth and output diversity.
AI vs. The Erdős Conjecture: A Parallel Computation Experiment
The intersection of Large Language Models (LLMs) and pure mathematics has entered a provocative new phase, as evidenced by the attempt to solve 20 of Paul Erdős's mathematical problems using 20 parallel OpenAI Codex accounts. This experiment represents a shift from using AI as a simple chatbot to utilizing it as a distributed computational engine. By leveraging multiple instances of Codex—the model that forms the backbone of GitHub Copilot—the project seeks to overcome the inherent stochastic limitations of single-prompt AI generations, attempting to find a path to a mathematical proof through sheer iterative volume and parallel exploration.
The Strategy of Parallelism and Ensemble Outputs
The decision to use 20 separate accounts running in parallel is a strategic move to combat the 'hallucination' and 'local minima' problems common in LLMs. In mathematical reasoning, a single wrong step in a proof renders the entire result invalid. By running multiple instances of Codex simultaneously, the researcher can generate a diverse ensemble of potential solutions. This approach mimics a 'Monte Carlo' method of reasoning, where the goal is to cast a wide net of potential logical paths, increasing the probability that at least one instance hits upon the correct sequence of operations or the necessary insight to solve a specific Erdős problem.
The Legacy of Paul Erdős and Mathematical Rigor
To understand the scale of this ambition, one must consider the nature of Paul Erdős's work. Erdős was one of the most prolific mathematicians of the 20th century, specializing in combinatorics, graph theory, and number theory. His problems are notorious for being deceptively simple to state but incredibly difficult to prove. Solving an Erdős problem typically requires not just computational power, but a 'leap' of intuition—a creative synthesis of mathematical concepts. By applying Codex to these problems, the experiment tests whether AI can simulate this intuition or if it can find a brute-force computational shortcut that human mathematicians have previously overlooked.
Codex: Bridging the Gap Between Code and Proof
OpenAI's Codex is uniquely suited for this task because mathematics and programming share a fundamental structural logic. Many complex mathematical proofs can be translated into formal languages or verified via code. By using Codex, the researcher is essentially treating the mathematical problems as coding challenges. If the AI can generate a script that proves a theorem or a program that verifies a conjecture for a vast set of numbers, the 'solution' becomes a tangible piece of software. This highlights a growing trend in 'Formal Verification,' where the goal is to move away from natural language proofs toward machine-verifiable logic.
Challenges in LLM-Based Mathematical Reasoning
Despite the ingenuity of the parallel approach, the experiment underscores the persistent struggle LLMs face with 'System 2' thinking—slow, deliberate, and logical reasoning. While Codex is excellent at pattern recognition and syntax, it lacks a native internal model of mathematical truth. It predicts the next most likely token based on training data rather than deriving a conclusion from first principles. The reliance on 20 parallel accounts is an admission that the model's reliability is low, necessitating a redundant system to filter the signal from the noise.
Future Implications for AI-Driven Science
This experiment is a precursor to a broader movement known as AI4Science. As we move toward more agentic AI workflows, we will likely see the rise of 'automated mathematician' frameworks that don't just guess answers but iterate, test, and self-correct in a loop. The use of parallel accounts is a primitive version of what will eventually become multi-agent orchestration, where different AI agents act as the 'prover' and the 'verifier,' debating a problem until a logically sound proof is reached. This could eventually lead to the resolution of long-standing conjectures that have eluded human genius for decades.
Summary: The attempt to solve Erdős problems using parallel Codex accounts is a fascinating study in the application of distributed AI to high-level mathematics. While it highlights the current limitations of LLM reliability, it also points toward a future where AI is used not as a replacement for the mathematician, but as a massive, parallelized search engine for mathematical truth.