# Réponse finale obtenue du bot:
response = « A fascinating topic! Let’s dive into the world of number theory and explore some of its most interesting properties, theorems, and conjectures.
**Properties of Integers**
1. **Commutativity**: The sum and product of two integers are independent of their order.
2. **Associativity**: The sum and product of three or more integers can be performed in any order.
3. **Distributivity**: The product of an integer and the sum of two integers is equal to the sum of its products with each integer separately.
**Theorems**
1. **Fermat’s Little Theorem**: If `p` is a prime number, then for any integer `a`, `a^(p-1) ≡ 1 (mod p)`.
2. **Euclid’s Lemma**: If `a` and `b` are integers such that `a | b*c` and `gcd(a, b) = 1`, then `a | c`.
3. **Euler’s Theorem**: For any integer `n` and `a` relatively prime to `n`, `(a^(φ(n))) ≡ 1 (mod n)`, where `φ(n)` is Euler’s totient function.
**Conjectures**
1. **Riemann Hypothesis**: A conjecture about the distribution of prime numbers, which has far-reaching implications for many areas of mathematics.
2. **Goldbach’s Conjecture**: Every even integer greater than 2 can be expressed as the sum of two prime numbers.
3. **Twin Prime Conjecture**: There are infinitely many pairs of primes that differ by 2.
**Cryptography**
1. **Public-Key Cryptography**: Based on the difficulty of factoring large composite numbers, public-key cryptosystems like RSA and elliptic curve cryptography provide secure data encryption and decryption.
2. **Hash Functions**: Used to ensure data integrity and authenticity, hash functions like SHA-256 and MD5 are built on number theoretical concepts.
Some famous results in number theory include:
* **The Prime Number Theorem**: Describes the distribution of prime numbers among the integers.
* **The Mordell-Weil Theorem**: States that for a given elliptic curve, the group of rational points is finitely generated.
* **The Modularity Theorem**: Establishes a connection between elliptic curves and modular forms.
These are just a few examples of the many fascinating properties, theorems, and conjectures in number theory. I hope this gives you a sense of the richness and beauty of this field! »