A **totative** is an integer that is coprime to a given integer \( n \).

The Tonelli–Shanks algorithm is a method used to compute square roots in finite fields, particularly useful for finding square roots of a number modulo a prime. This algorithm is significant in number theory and has applications in cryptography, especially in schemes dealing with quadratic residues.

Thue's lemma, also known as Thue's theorem, is a result in the field of Diophantine approximation and number theory, named after the mathematician Axel Thue. The lemma addresses the approximation of real numbers by rationals and is particularly concerned with the properties of certain algebraic numbers.

A table of congruences is a systematic way to present the relationships between integers under modular arithmetic. It displays which numbers are congruent to each other modulo a particular base (or modulus). In modular arithmetic, two integers \( a \) and \( b \) are said to be congruent modulo \( n \) (written as \( a \equiv b \mod n \)) if they have the same remainder when divided by \( n \).

The Solovay–Strassen primality test is a probabilistic algorithm used to determine whether a given number is prime. It was developed independently by Robert Solovay and Jeffrey Strassen in the early 1970s. The test is based on properties of quadratic residues and the law of quadratic reciprocity. ### How the Test Works 1. **Input**: The algorithm takes an odd positive integer \( n \) greater than 1.

A **root of unity** modulo \( n \) refers to an integer \( k \) such that \( k^m \equiv 1 \mod n \) for some positive integer \( m \). In other words, \( k \) is a root of unity if it raises to some integer power \( m \) and gives a result of 1 when taken modulo \( n \).

The Residue Number System (RNS) is a non-weighted number system used in digital computation and signal processing that represents integers using their residues with respect to a set of pairwise coprime moduli. It provides several advantages such as high parallelism, reduced carry propagation, and potential speed improvements in arithmetic operations.

A **reduced residue system** is a set of integers that are representatives of the distinct equivalence classes of integers modulo \( n \), where \( n \) is a positive integer, and each representative in the set is coprime to \( n \). In other words, a reduced residue system modulo \( n \) consists of integers that are both less than \( n \) and relatively prime to \( n \).

Quartic reciprocity is a concept in number theory that extends the ideas of quadratic reciprocity to higher powers, specifically to quartic residues. Just as quadratic reciprocity provides conditions under which two primes can be classified as quadratic residues or non-residues, quartic reciprocity deals with congruences of the form \(x^4 \equiv a \mod p\).

A quadratic residue is a concept from number theory, particularly in the study of modular arithmetic.

Quadratic reciprocity is a fundamental theorem in number theory that describes the solvability of quadratic equations modulo prime numbers. It addresses the question of when a quadratic residue exists for two distinct odd prime numbers.

Fermat's Little Theorem is a fundamental result in number theory that states: If \( p \) is a prime number and \( a \) is any integer not divisible by \( p \), then: \[ a^{p-1} \equiv 1 \mod p \] This means that when \( a^{p-1} \) is divided by \( p \), the remainder is 1.

A primitive root modulo \( n \) is an integer \( g \) such that its powers generate all the integers coprime to \( n \) up to \( n \).

A polydivisible number is a number that meets a specific divisibility condition related to its digits. Specifically, a positive integer is considered polydivisible if for every \( k \) (where \( k \) is the position of the digit from the left), the number formed by the first \( k \) digits is divisible by \( k \).

Pocklington's algorithm is a method used to test the primality of large integers. It was developed by the mathematician Henry Pocklington in 1914 and is particularly effective for numbers that can be represented in a specific form. The algorithm is based on the properties of prime numbers and relies on certain mathematical theorems related to divisibility and modular arithmetic.

The Pisano period, denoted as \( \pi(m) \), is the period with which the sequence of Fibonacci numbers repeats modulo \( m \). In other words, if you take the Fibonacci sequence \( F_0, F_1, F_2, \ldots \), and reduce each number modulo \( m \), the resulting sequence will eventually start repeating. The length of this repeating sequence is known as the Pisano period for \( m \).

The multiplicative order of an integer \( a \) modulo \( n \) is defined as the smallest positive integer \( k \) such that \[ a^k \equiv 1 \mod n. \] In simpler terms, it is the smallest exponent \( k \) for which raising \( a \) to the power of \( k \) results in a value that, when divided by \( n \), leaves a remainder of 1.

Modulo, often represented by the symbol `%`, is a mathematical operation that finds the remainder of the division of one integer by another.

The Method of Successive Substitution is a technique used to solve equations, particularly in the context of finding fixed points of functions or solutions of nonlinear equations. The essence of the method is to iteratively approximate a solution by repeatedly substituting back into the original equation until a satisfactory level of accuracy is reached. ### Steps in the Method of Successive Substitution: 1. **Rearrangement**: The original equation is rearranged into a form that isolates one variable.

The Legendre symbol is a mathematical notation that provides a way to determine if a given integer is a quadratic residue modulo a prime number. Specifically, for an integer \( a \) and a prime \( p \), the Legendre symbol is denoted as: \[ \left( \frac{a}{p} \right) \] It is defined as follows: 1. If \( a \) is congruent to 0 modulo \( p \) (i.e.