Congruence

Modular arithmetic: This is the foundation of studying congruence in number theory. It is a way of working with integers in a limited range by considering only the remainder when divided by a fixed positive integer.

Fermat's Little Theorem: This theorem states that if p is a prime number and a is an integer not divisible by p, then a^p-1 ≡ 1 mod p. This is useful in proving properties of congruences.

Euler's Totient Function: This function counts the number of positive integers less than n that are relatively prime to n. It has many applications in number theory, including calculating modular inverses.

Chinese Remainder Theorem: This theorem states that if m and n are relatively prime positive integers, then any two numbers modulo mn can be uniquely represented as a pair of numbers modulo m and modulo n, respectively.

Wilson's Theorem: This theorem states that a positive integer p is prime if and only if (p-1)! ≡ -1 mod p. It is useful in checking primality of large numbers.

Primitive Roots: A primitive root modulo n is an integer a such that every integer relatively prime to n can be written as a power of a modulo n. They have many applications in cryptography and number theory.

Quadratic Residues: An integer a is called a quadratic residue modulo n if there exists an integer x such that x^2 ≡ a mod n. This topic has applications in cryptography and coding theory.

Quadratic Reciprocity: This theorem expresses a necessary and sufficient condition for the solvability of certain quadratic congruences. It has many applications in number theory and cryptography.

Diffie-Hellman Key Exchange: This is a cryptographic protocol that allows two parties to agree on a shared secret key over an insecure communication channel. It is based on the discrete logarithm problem in finite fields.

Elliptic Curve Cryptography: This is a modern encryption system based on the arithmetic of elliptic curves. It is widely used in secure communication protocols, such as HTTPS and PGP.

Modulo Congruence: A ≡ B (mod m): Two integers A and B are said to be congruent modulo m if and only if their difference (A-B) is divisible by m.

Geometric Congruence: Two geometric figures are said to be congruent if they have the same shape and size.

Group Congruence: When two groups are structurally identical, we say that they are isomorphic or congruent. This is a concept from abstract algebra.

Quadratic Congruence: An equation of the form ax^2 + bx + c ≡ 0 (mod m), where a, b, and c are integers, and m is a positive integer, is known as a quadratic congruence.

Modular Inverse Congruence: An integer A is said to have an inverse modulo m if there exists some integer B such that AB ≡ 1 (mod m). This is called the modular inverse of A modulo m.

Linear Congruence: A linear congruence is an equation of the form ax ≡ b (mod m), where a, b, and m are integers, and a and m are coprime.

Hensel's Lemma Congruence: Hensel's lemma is a tool in number theory that helps solve certain types of congruences. It deals with lifting a solution of a polynomial equation modulo p to a solution modulo p^2, p^3, and so on.

Lucas's Theorem Congruence: Lucas's theorem is a result in number theory that helps evaluate binomial coefficients modulo a prime p. It is important in combinatorics and cryptography.

Chinese Remainder Theorem Congruence: The Chinese Remainder Theorem is a fundamental result in number theory that allows us to find a unique solution to a system of congruences.

Wilson Congruence: The Wilson Congruence is an equation in number theory that states that (p-1)! ≡ -1 (mod p) for a prime p.