# Réponse finale obtenue du bot:
response = « What a fascinating topic! Let’s dive into the world of number theory and explore some of its most iconic properties, theorems, and conjectures.
**Properties of Integers**
1. **Divisibility**: Every integer can be uniquely expressed as a product of prime numbers (Fundamental Theorem of Arithmetic).
2. **Prime Numbers**: There are infinitely many prime numbers (Euclid’s proof).
3. **Fermat’s Little Theorem**: For any prime number p and integer a, $a^{p-1} \equiv 1 \mod p$.
4. **Wilson’s Theorem**: For any prime number p, $(p-1)! \equiv -1 \mod p$.
**Theorems**
1. **Fermat’s Last Theorem (FLT)**: There are no integer solutions to the equation $a^n + b^n = c^n$ for $n > 2$. (Proved by Andrew Wiles in 1994).
2. **Riemann Hypothesis**: All non-trivial zeros of the Riemann zeta function lie on a vertical line in the complex plane.
3. **Twin Prime Conjecture**: There are infinitely many pairs of prime numbers that differ by 2 (e.g., 3 and 5, or 11 and 13).
4. **Goldbach’s Conjecture**: Every even integer greater than 2 can be expressed as the sum of two prime numbers.
**Conjectures**
1. **Riemann Hypothesis** (mentioned earlier): This conjecture has far-reaching implications for many areas of mathematics, including number theory and cryptography.
2. **Twin Prime Conjecture** (mentioned earlier): Despite much effort, this conjecture remains unsolved.
3. **Modular Form Conjecture**: The set of modular forms is a fundamental object in number theory, but its properties are still not fully understood.
**Cryptographic Applications**
1. **Public-Key Cryptography**: Many encryption algorithms rely on the difficulty of factoring large integers (e.g., RSA) or computing discrete logarithms (e.g., Diffie-Hellman).
2. **Hash Functions**: Cryptographic hash functions, like SHA-256, use number-theoretic properties to ensure data integrity and authenticity.
3. **Random Number Generation**: Pseudorandom number generators often rely on the properties of prime numbers and modular arithmetic.
This is just a brief overview of the fascinating world of number theory. I hope this has piqued your interest in this field! »