1. Proof of Fermat's Last Theorem: Fermat's Last Theorem is a mystery of a century. This theorem was finally proved by andrew wiles in 1995. He proved this theorem by "infinite descent method" in number theory, which is considered as one of the greatest proofs in mathematics.

2. Proof of matrix multiplication: Although matrix multiplication is simple and easy to understand, it is a very important mathematical concept and widely used in computer science and engineering. The proof of matrix multiplication is also very interesting, which involves matrix operation and vector space theory, and also needs some abstract mathematical concepts.

3. Proof of mean inequality: Mean inequality is a basic inequality, which has been applied in many fields. It claims that the proof of this theorem involves mathematical induction and inequality theory for positive real numbers, but it is very beautiful and concise.

4. Proof of Fermat's Last Theorem: Fermat's Last Theorem is a simple and practical algorithm for testing prime numbers. It claims that for prime numbers and arbitrary integers, the proof of this theorem can be carried out by modular operation and euler theorem.

5. Proof of Euler formula: Euler formula is one of the most beautiful and mysterious formulas in mathematics, which describes the relationship between three basic mathematical constants. The proof of Euler formula needs the theory of series and complex number, but it is very beautiful and wonderful.

In short: there are many wonderful proofs in mathematics, which is the charm of mathematics!