Public-key cryptography, also known as asymmetric cryptography, is a cryptographic system that uses a pair of keys, a public key and a private key, to encrypt and decrypt messages.

Primality testing, integer factorization, and finding the greatest common divisor are all problems that are frequently encountered in computer science and are often solved using techniques from number theory.

Pythagorean tuning is a system of musical tuning based on the mathematical relationship between the notes in a musical scale. It is named after the Greek philosopher Pythagoras, who is credited with discovering the mathematical basis of music.

The RSA algorithm, Diffie-Hellman key exchange, and elliptic curve cryptography are all examples of number-theoretic algorithms that are widely used in practice for secure communication and data protection.

The fundamental theorem of arithmetic states that every positive integer can be written as a unique product of prime numbers. This theorem is a cornerstone of number theory and has many applications in various fields.

Number theory plays a crucial role in coding theory, which is concerned with the efficient transmission and storage of information. Applications of number theory in coding theory include error-correcting codes, data compression, and network coding.

In public-key cryptography, the modulus is a large prime number that is used as part of the public key. The modulus is used to encrypt messages and verify digital signatures.

Goldbach's conjecture, twin prime conjecture, and Collatz conjecture are all examples of number-theoretic problems that have been studied extensively but remain unsolved. These problems continue to challenge mathematicians and attract significant research efforts.

Primality testing algorithms, integer factorization algorithms, and random number generators are all examples of algorithms that are based on number theory and have practical applications in computer science.

Number theory has applications in various areas of physics, including quantum cryptography, number theory in string theory, and number theory in cosmology. These applications demonstrate the interdisciplinary nature of number theory and its relevance to diverse fields of study.

Euler's totient function, denoted by φ(n), is a function that counts the number of positive integers less than or equal to n that are relatively prime to n.

Number theory has applications in various areas of finance, including pricing financial derivatives, risk management, and portfolio optimization. These applications demonstrate the practical relevance of number theory in the financial world.

Dirichlet's theorem on primes in arithmetic progressions states that for any positive integer n, there are infinitely many primes p such that p ≡ 1 (mod n). This theorem has important applications in number theory and its subfields.

Number theory has applications in combinatorial game theory, which is a branch of game theory that studies games with perfect information and no chance moves. Number-theoretic concepts such as nim-sums and Sprague-Grundy functions are used to analyze and solve combinatorial games.

The partition function, denoted by p(n), gives the number of ways to represent a positive integer n as a sum of distinct positive integers. It is a classical function in number theory with a rich history and many applications.