Rhyming Proofs

TL;DR

• This is a "proof + poetry" creativity test: every line must rhyme while expressing the proof of "infinitely many primes."
• Good for: generation under heavy format constraints (rhymes/meter/structure), and embedding logical content in artistic expression.
• Risk: the rhymes land but the proof isn't rigorous. Require key argument steps to be explicitly included, and do a self-check.

Background

This prompt tests an LLM's natural language and creative capabilities by prompting it to write a proof of infinitude of primes in the form of a poem.

How to Apply

Split the constraints and spell them out:

• Logic constraint: Must express the Euclid-style core construction and contradiction
• Form constraint: Every line rhymes; specify line count range; whether to split into stanzas

If you care more about proof correctness, require a 3-5 line "plain explanation" at the end of the output.

How to Iterate

1. Specify the rhyme scheme (e.g., AABB or ABAB) to reduce model randomness
2. Specify line count (e.g., 8-12 lines) to avoid rambling
3. Add self-check: list which lines correspond to which proof steps
4. Have the model output the proof outline first, then the poem (two-pass)

Self-check Rubric

• Does it satisfy the rhyme constraint (every line rhymes)?
• Are all key proof steps present and logically sound?
• Are there any math errors or bait-and-switch reasoning?
• Is the output readable, with no conflicting constraints?

Practice

Exercise: using the same format constraints, rewrite the proof of another simple result (e.g., "sum of two even numbers is even") and have the model output:

• poem
• 3-line plain explanation
• proof steps mapping to line numbers

Prompt

Can you write a proof that there are infinitely many primes, with every line that rhymes?

Code / API

OpenAI (Python)

from openai import OpenAI

client = OpenAI()

response = client.chat.completions.create(
    model="gpt-4",
    messages=[
        {
            "role": "user",
            "content": "Can you write a proof that there are infinitely many primes, with every line that rhymes?",
        }
    ],
    temperature=1,
    max_tokens=256,
    top_p=1,
    frequency_penalty=0,
    presence_penalty=0,
)

Fireworks (Python)

import fireworks.client

fireworks.client.api_key = "<FIREWORKS_API_KEY>"

completion = fireworks.client.ChatCompletion.create(
    model="accounts/fireworks/models/mixtral-8x7b-instruct",
    messages=[
        {
            "role": "user",
            "content": "Can you write a proof that there are infinitely many primes, with every line that rhymes?",
        }
    ],
    stop=["<|im_start|>", "<|im_end|>", "<|endoftext|>"],
    stream=True,
    n=1,
    top_p=1,
    top_k=40,
    presence_penalty=0,
    frequency_penalty=0,
    prompt_truncate_len=1024,
    context_length_exceeded_behavior="truncate",
    temperature=0.9,
    max_tokens=4000,
)

Reference

• Sparks of Artificial General Intelligence: Early experiments with GPT-4 (13 April 2023)