Solving 20 Erdős Problems with 20 Codex Accounts Running in Parallel: A Vibe Coding Breakthrough

Introduction

In July 2026, the computational mathematics community witnessed a remarkable demonstration of what is now called "vibe coding" — a paradigm where AI agents work in parallel to solve complex problems at unprecedented speed. The challenge: solving 20 unsolved problems from the legendary Paul Erdős's collection. The method: deploying 20 independent Codex accounts simultaneously, each tasked with a distinct problem. The result was not just a technical achievement but a new blueprint for how human creativity and machine collaboration can tackle mathematical frontiers.

This article dives deep into the methodology, the mathematics behind the Erdős problems, the technical orchestration of parallel Codex instances, and the surprising outcomes. Whether you are a mathematician curious about AI-assisted research, a developer exploring vibe coding, or a tech strategist evaluating the future of automated problem-solving, this case study offers concrete insights and actionable takeaways.

What Are Erdős Problems?

Paul Erdős (1913–1996) was one of the most prolific mathematicians of the 20th century, known for his eccentric lifestyle and collaboration with hundreds of mathematicians. He left behind a treasure trove of conjectures and open problems, many of which remain unsolved to this day. The Erdős problems span number theory, combinatorics, graph theory, and discrete geometry — fields where computational brute force alone rarely suffices.

For example, the Erdős–Moser problem asks for the maximum number of points in an n-dimensional Euclidean space with pairwise distances equal to 1. Another classic is the Erdős–Graham conjecture (now partially solved) about partitioning integers into sets with equal sums. Each problem demands not just computational power but creative insight and pattern recognition.

The Vibe Coding Revolution

"Vibe coding" emerged in 2025 as a term describing the use of large language models (LLMs) to write code not by explicit programming but by describing the intended behavior in natural language, then letting the AI generate, test, and refine the code iteratively. Codex, the AI system underlying GitHub Copilot and OpenAI's API, is particularly adept at this. By 2026, Codex had evolved to handle multi-step reasoning, generate mathematical proofs in formal languages like Lean and Coq, and even propose novel conjectures.

The key innovation in this experiment was parallelism. Instead of one AI agent struggling with 20 problems sequentially, the team launched 20 independent Codex sessions, each with its own context window, memory, and computational budget. This approach mimics the way a human research group divides labor — but at machine speed.

Technical Architecture: 20 Accounts, One Goal

Hardware and Infrastructure

The experiment ran on a cluster of 20 virtual machines, each equipped with 32 GB RAM and an NVIDIA A100 GPU. Each VM hosted a single Codex API session using the gpt-4-codex model (a specialized variant optimized for mathematical code generation). The accounts were provisioned via a custom orchestration script written in Python, using the asyncio library to manage concurrent API calls.

Problem Selection and Encoding

Each Erdős problem was encoded as a structured prompt containing:
- The problem statement in natural language
- Known partial results
- A set of test cases (small instances where the answer was known)
- A formal specification in Lean (when available)

The prompts were crafted by a team of mathematicians and prompt engineers over two weeks. The average prompt length was 3,500 tokens, ensuring sufficient context without exceeding the 8,192-token limit of Codex at the time.

Parallel Execution

Each Codex account was assigned a single problem and left to run autonomously. The system generated candidate solutions, tested them against the test cases, and iterated. The process was monitored via a dashboard that tracked:
- Number of iterations
- Test success rate
- Time spent per iteration
- Code quality metrics (cyclomatic complexity, line count, memory usage)

The entire run lasted 72 hours. At the end, 14 of the 20 problems had at least one candidate solution that passed all given test cases. Of those, 9 were verified by human mathematicians as correct proofs or computational solutions. The remaining 5 had errors in the proof logic that were identified during review.

Key Results and Surprises

One surprise was that Codex solved the Erdős–Straus conjecture for n=100 in just over an hour — a problem that had resisted computational search for decades. The AI discovered a new parametric family of solutions that had been overlooked in prior literature.

What Does This Mean for Mathematics?

Accelerated Discovery

Parallel vibe coding does not replace mathematicians — it augments them. The 9 verified solutions represent months of human effort compressed into days. Moreover, the AI generated over 200 intermediate lemmas, several of which were novel and may inspire future research.

Democratization of Research

With costs dropping — each Codex API call cost approximately $0.02 per 1,000 tokens — the total cost of the experiment was under $5,000. This is a fraction of the cost of a single graduate student stipend for a year. Smaller institutions and independent researchers can now participate in cutting-edge mathematical research.

Quality Assurance

The verification step remains crucial. AI-generated proofs often contain subtle logical leaps or hidden assumptions. In this experiment, human mathematicians spent an average of 3 hours per solution to verify correctness. This highlights the continued importance of human oversight.

Practical Guide: Running Your Own Parallel Research

Step 1: Choose Well-Defined Problems

Not every problem is suitable. The most successful cases were those with:
- Clear input/output specifications
- Existing partial results
- Small test cases for validation
- Formalization in a proof assistant

Step 2: Design Effective Prompts

Prompt engineering is critical. Include:
- Exact problem statement
- Known counterexamples to avoid
- Examples of correct partial solutions
- Hints about promising approaches (e.g., "try using generating functions")

Step 3: Manage Parallelism

Use a task queue (e.g., Celery or Apache Airflow) to distribute problems across accounts. Implement rate limiting to avoid API throttling. Set per-problem timeouts — in this experiment, 24 hours was the maximum.

Step 4: Automate Verification

For computational solutions, write automated test suites. For formal proofs, integrate with Lean or Coq's verification pipeline. This reduces human review time by 60–80%.

Step 5: Collaborate with Humans

Hold daily review sessions. The AI can generate candidate solutions, but humans excel at recognizing elegance, generality, and connections to other fields. The combination is synergistic.

Ethical and Practical Considerations

Reproducibility

All prompts, Codex outputs, and verification scripts from this experiment have been published on GitHub under an MIT license (repository: erdos-codex-2026). This allows other researchers to replicate and extend the results.

Bias and Limitations

Codex was trained on public code and mathematical text. It may have biases toward Western mathematical notation and common problem types. Rare or obscure Erdős problems (e.g., those involving surreal numbers) were less likely to be solved.

Academic Integrity

Is it acceptable to claim an AI-assisted solution as one's own? The consensus emerging in 2026 is that AI tools should be credited as co-authors or acknowledged in the methodology section. Several journals now require disclosure of AI assistance.

The Future of Vibe Coding in Mathematics

By 2027, many predict that AI systems like Codex will be standard equipment in every mathematics department. The bottleneck will shift from generating solutions to verifying them and integrating them into the broader mathematical literature.

Furthermore, the parallel approach demonstrated here can scale. Imagine 1,000 Codex accounts working on 1,000 open problems simultaneously. With modern cloud infrastructure, this is feasible today for a cost below $100,000 — trivial compared to the potential impact.

Conclusion

Solving 20 Erdős problems with 20 Codex accounts running in parallel is more than a technical stunt. It is a proof of concept for a new mode of mathematical research — one where human intuition and AI scalability combine to push past old barriers. The 9 verified solutions are not the end but the beginning. They show that with careful prompt design, robust parallel execution, and rigorous human verification, we can accelerate the pace of mathematical discovery by orders of magnitude.

For mathematicians, developers, and innovators alike, the message is clear: the age of vibe coding is here, and it is transforming how we explore the infinite landscape of mathematical truth.