Infinite primes (Shakespeare)

creative proof prompt example

#TL;DR（中文）

这是一个 “数学证明 + 风格写作” 的
code
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
code
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：

Content constraints：必须包含哪些核心论证步骤（proof skeleton）
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）

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

markdown
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)

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)

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)

Prompt 大师