Please use deep thinking mode to answer the following questions:




Prove: For any positive integer n, n³ - n is divisible by 6.

I need to choose a caching solution for a web application with 100,000
daily users. Please compare and analyze:




Option A: Redis cluster


Option B: Local memory cache (such as Caffeine)




Analyze and give recommendations from the four dimensions of performance, cost,
operation and maintenance complexity, and data consistency.

Please answer LeetCode question 146 LRU cache, requirements:


1. Give Python implementation


2. Time complexity O(1)


3. Explain in detail the reasons for choosing the data structure


4. Analyze boundary conditions

There is the following table structure:


- orders(id, user_id, amount, created_at, status)


- users(id, name, email, created_at)




Please write a SQL query: find the users whose total consumption exceeds 1,000
yuan in the past 30 days.


Sorted in descending order by total consumption, display user name, email address,
number of orders, and total consumption.




At the same time, analyze the performance bottleneck of this query and recommend
how to add indexes.

Please write a technical blog with the title "Understanding Transformer
Attention Mechanism from Scratch" for readers who have basic Python but no deep
learning background.




Requirements:


- Explain the attention mechanism using metaphors from life


- Contains simple explanations of mathematical formulas (just use text descriptions)


- With pseudocode examples


- Approximately 800 words in length


- Markdown format