Problem
For decades, mathematicians have wrestled with the unit distance problem—a question that asks how many pairs of points at exactly one unit apart can exist in a planar set. The problem sits at the heart of discrete geometry and has resisted resolution for roughly 80 years. Without a reliable computational partner, researchers must rely on hand‑crafted arguments, which are time‑consuming and error‑prone.
According to the OpenAI Blog, an OpenAI model recently solved this long‑standing problem and, in doing so, disproved a major conjecture in the field. The breakthrough demonstrates that AI can generate rigorous mathematical reasoning at a scale previously unseen. However, the model’s success raises a new question: how can other scholars tap into this capability for their own geometric conjectures?
Prerequisites
Before you start, make sure you have the following:
- A valid OpenAI API key that includes access to the research‑oriented model referenced in the blog post.
- Familiarity with the basics of discrete geometry, especially concepts like point sets, distances, and graph embeddings.
- Python (or another supported language) installed on your workstation, along with the OpenAI SDK.
- A version‑controlled repository (Git) to track prompts, responses, and revisions.
- Optional: A symbolic computation library such as SymPy for post‑processing the model’s output.
Steps
1. Set Up Your Development Environment
Install the OpenAI Python package and verify your API key works. A quick test call—asking the model to state the unit distance problem—confirms connectivity.
2. Formalize the Conjecture
Translate your geometric question into a precise, machine‑readable statement. Include definitions, constraints, and any known lemmas. For the unit distance problem, the model was fed a description of point sets in the Euclidean plane and the distance condition.
3. Craft a Prompt That Guides Reasoning
Structure the prompt in three parts: (a) context, (b) task, and (c) format. Example:
Context: You are a mathematician specializing in discrete geometry. The unit distance problem asks for the maximal number of unit‑distance pairs in a set of n points in the plane. Task: Provide a proof or counterexample to the conjecture that the maximal number grows linearly with n. Format: Return a step‑by‑step logical argument, ending with a clear conclusion.
This style mirrors the prompt that led to the breakthrough, as described by OpenAI.
4. Submit the Prompt and Retrieve the Response
Use the SDK’s chat.completions.create method, setting temperature to a low value (e.g., 0.2) to favor deterministic reasoning. Capture the raw text and store it alongside the prompt.
5. Parse the Output
Extract the logical steps, equations, and any constructed point configurations. If the model supplies a diagram description, translate it into coordinates you can plot with Matplotlib.
6. Verify the Argument
Independent verification is essential. Feed each logical step back to the model as a separate query: “Is step 3 valid given steps 1‑2?” Combine the model’s self‑checks with traditional proof‑checking tools.
7. Iterate and Refine
If gaps appear, adjust the prompt—add missing lemmas, tighten constraints, or request alternative approaches. The model’s ability to explore multiple proof pathways was key to the unit distance result.
8. Document the Process
Commit the final prompt, response, and verification notes to your repository. Include a README that explains how the model was used, so peers can reproduce the result.
Pro Tips
- Start Small. Test the model on known theorems before tackling open conjectures. Success on familiar ground builds confidence.
- Use Structured Output. Ask the model to return JSON or LaTeX fragments. Structured data simplifies downstream checks.
- Combine With Symbolic Engines. Feed algebraic results into SymPy to automatically simplify or detect contradictions.
- Limit Token Length. Long prompts can dilute focus. Keep the core problem under 1,000 tokens; use follow‑up calls for extensions.
- Maintain a Prompt Library. Archive effective prompt templates. The unit distance breakthrough stemmed from an iterative prompt‑engineering process.
The OpenAI model’s success on the unit distance problem proves that AI can generate rigorous, novel mathematics. By following this guide, you can bring that capability into your own research workflow.
