# Réponse finale obtenue du bot:
response = « A fascinating topic! Here’s an exploration of the properties of numbers and their applications in number theory:
**Properties of Integers**
1. **Commutativity**: The order in which integers are added or multiplied does not change the result.
2. **Associativity**: The order in which integers are added or multiplied can be changed without affecting the result.
3. **Distributivity**: Multiplication distributes over addition, meaning that `a × (b + c) = a × b + a × c`.
4. **Closure under addition and multiplication**: The result of adding or multiplying two integers is always an integer.
**Theorems and Conjectures**
1. **Fermat’s Last Theorem (FLT)**: A theorem that states there are no integer solutions to `a^n + b^n = c^n` for `n > 2`.
2. **Riemann Hypothesis**: A conjecture about the distribution of prime numbers, stating that all non-trivial zeros of the Riemann zeta function lie on a vertical line in the complex plane.
3. **Goldbach’s Conjecture**: A conjecture that every even integer greater than 2 can be expressed as the sum of two prime numbers.
**Cryptography**
1. **Public-key cryptography**: Uses large integers and modular arithmetic to ensure secure data transmission. Examples include RSA (Rivest-Shamir-Adleman) and elliptic curve cryptography.
2. **Hash functions**: Use integer properties, such as modulo operations, to create a digital fingerprint of data.
**Applications**
1. **Computer Science**: Number theory is used in algorithms for tasks like encryption, error-correcting codes, and random number generation.
2. **Cryptography**: As mentioned earlier, number theory is the foundation of public-key cryptography.
3. **Mathematical Physics**: Number theory appears in the study of quantum mechanics, particularly in the context of prime numbers and modular forms.
This is just a brief overview of the fascinating world of number theory. I hope you found it helpful! »