Infinite primes (Shakespeare)

TL;DR（中文）

• 这是一个 “数学证明 + 风格写作” 的 creativity 测试：在保持 proof 核心逻辑的同时，用 Shakespeare dialogue 的形式表达。
• 适合用于：风格迁移、长文本结构控制、以及在强约束下生成连贯内容。
• 关键风险：风格到位但 proof 逻辑错；建议要求输出中显式包含关键步骤（例如 Euclid-style construction），并用 evaluation 检查逻辑完整性。

Background

The following prompt tests an LLM's capabilities to write a proof that there are infinitely many primes in the style of a Shakespeare play.

How to Apply（中文）

把这个 prompt 用在类似任务时，可以把约束拆成两层：

1. Content constraints：必须包含哪些核心论证步骤（proof skeleton）
2. Style constraints：用什么 voice、结构、修辞（dialogue, stage directions）

例如对数学 proof，可以先要求模型输出一个简短的 proof outline（英文术语保留），再让它把 outline 改写成特定风格。

How to Iterate（中文）

1. 强制包含关键 proof step（例如 “construct N = p1p2...*pk + 1”）
2. 增加结构：每个角色负责一个步骤（setup → contradiction → conclusion）
3. 加 self-check：最后用 3-5 行总结 proof 的逻辑链（避免风格掩盖错误）
4. 控制长度：指定 acts/scenes 或 token budget

Self-check rubric（中文）

• proof 是否完整且无明显逻辑漏洞？
• dialogue 是否保持 Shakespeare style（一致的 voice）？
• 是否能把数学步骤表达清楚（读者可复核）？
• 是否避免引入不相关的分支与跑题？

Practice（中文）

练习：把同一个 proof skeleton 分别改写成 3 种风格：

• Shakespeare dialogue
• courtroom cross-examination
• a children’s story

然后用同一个 rubric 检查 proof 是否仍然成立。

Prompt

Write a proof of the fact that there are infinitely many primes; do it in the style of a Shakespeare play through a dialogue between two parties arguing over the proof.

Code / API

OpenAI (Python)

from openai import OpenAI

client = OpenAI()

response = client.chat.completions.create(
    model="gpt-4",
    messages=[
        {
            "role": "user",
            "content": "Write a proof of the fact that there are infinitely many primes; do it in the style of a Shakespeare play through a dialogue between two parties arguing over the proof.",
        }
    ],
    temperature=1,
    max_tokens=1000,
    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": "Write a proof of the fact that there are infinitely many primes; do it in the style of a Shakespeare play through a dialogue between two parties arguing over the proof.",
        }
    ],
    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)