Modular arithmetic, often referred to as "clock arithmetic," is a system of arithmetic for integers, where numbers wrap around after reaching a certain value known as the modulus. In modular arithmetic, two numbers are considered equivalent if they have the same remainder when divided by the modulus. The basic notation for modular arithmetic is \( a \equiv b \mod m \), which means that \( a \) and \( b \) give the same remainder when divided by \( m \).
An automorphic number is a number whose square ends with the same digits as the number itself. In other words, if \( n \) is an automorphic number, then when you compute \( n^2 \), the last digits of \( n^2 \) will be the same as \( n \). For example: - The number 5 is automorphic because \( 5^2 = 25 \), and the last digit (5) matches the original number.
Barrett reduction is an algorithm used in the field of modular arithmetic, particularly for efficiently reducing large integers modulo a smaller integer. It is especially useful in cryptography and in computations involving large numbers where performance is critical. The Barrett reduction method is designed to avoid the expensive division operation typically associated with modular reduction. Instead of performing a direct division, it leverages precomputed values to carry out the reduction more efficiently. ### Basic Steps of Barrett Reduction 1.
"Canon arithmeticus" is a term that refers to a work by the mathematician John Napier, published in 1614. The full title in Latin is "Mirifici Logarithmorum Canonis Descriptio." This work introduced and laid the groundwork for the concept of logarithms, which are critical in mathematics, particularly for simplifying calculations involving multiplication and division.
The Carmichael function, denoted as \( \lambda(n) \), is a function that gives the least positive integer \( m \) such that \( a^m \equiv 1 \pmod{n} \) for all integers \( a \) that are coprime to \( n \). In other words, it is the smallest exponent for which every number coprime to \( n \) has a power that is congruent to 1 modulo \( n \).
A Carmichael number is a composite number \( n \) that satisfies Fermat's little theorem for all integers \( a \) that are coprime to \( n \). Specifically, Fermat's little theorem states that if \( p \) is a prime number, then for any integer \( a \) such that \( a \) is not divisible by \( p \), \[ a^{p-1} \equiv 1 \ (\text{mod} \ p).
The Chinese Remainder Theorem (CRT) is a result in number theory that provides a way to solve systems of simultaneous congruences with different moduli. It states that if you have several congruences with pairwise coprime moduli, there exists a unique solution modulo the product of those moduli. ### Formulation If \( n_1, n_2, \ldots, n_k \) are pairwise coprime integers (i.e.
Cipolla's algorithm is a method used for efficiently computing square roots in finite fields, specifically quadratic residues, which is particularly useful in the context of cryptographic applications and certain areas of number theory. It is named after the mathematician Giovanni Cipolla.
Congruence of squares is a concept in number theory that deals with whether two numbers can be expressed as squares mod some integer. Specifically, it investigates under what conditions a quadratic residue (a number that is congruent to a perfect square modulo \( n \)) can be expressed as the square of another number modulo \( n \).
Cubic reciprocity is a concept in number theory similar to quadratic reciprocity, but it deals specifically with cubic residues and their properties. While quadratic reciprocity provides a criterion for determining whether a given integer is a quadratic residue modulo a prime, cubic reciprocity focuses on the behavior of cubic residues.
The term "discrete logarithm records" generally refers to records of algorithms, properties, or particular instances related to the discrete logarithm problem, which is a fundamental problem in number theory and cryptography.
Euler's criterion is a result in number theory that provides a way to determine whether a given integer is a quadratic residue modulo a prime number.
Euler's theorem is a fundamental statement in number theory that relates to modular arithmetic. It is particularly useful for working with integers and their properties under modular exponentiation. The theorem states that if \( a \) and \( n \) are coprime (i.e.
Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then the following congruence holds: \[ a^{p-1} \equiv 1 \mod p \] This means that when \( a^{p-1} \) is divided by \( p \), the remainder is 1.
In number theory, Gauss's lemma is a result that relates to the quadratic residues modulo a prime. Specifically, it provides a criterion for determining whether a given integer is a quadratic residue modulo a prime number. The statement of Gauss's lemma can be formalized as follows: Let \( p \) be an odd prime, and let \( a \) be an integer that is not divisible by \( p \).
The Jacobi symbol is a mathematical notation that generalizes the Legendre symbol. It is used primarily in number theory, particularly in the context of quadratic residues and the study of prime numbers.
Jordan's totient function, denoted as \( J_k(n) \), is a generalization of Euler's totient function.
Kronecker's congruence refers to a specific mathematical relationship concerning integer sequences and their congruences. In the context of number theory, particularly, it identifies conditions under which two sequences or sums are congruent modulo some integer. A classic representation of Kronecker's congruence is in the context of partition functions, where one often studies the congruences of partition numbers.
The Kronecker symbol, denoted as \(\left(\frac{a}{n}\right)\), is a generalization of the Legendre symbol used in number theory. It is defined for any integer \(a\) and any positive integer \(n\) that can be expressed as a product of prime powers. The Kronecker symbol extends the properties of the Legendre symbol to include not just odd prime moduli, but also powers of 2 and arbitrary positive integers.
Kummer's congruence is a result in number theory concerning the distribution of prime numbers in relation to binomial coefficients. Specifically, it addresses the behavior of binomial coefficients \( \binom{p}{k} \) modulo a prime \( p \).
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.
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.
Modulo, often represented by the symbol `%`, is a mathematical operation that finds the remainder of the division of one integer by another.
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.
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 \( m \).
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.
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 \( k \).
A primitive root modulo \( n \) is an integer \( g \) such that its powers generate all the integers coprime to \( n \) up to \( n \).
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.
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.
A quadratic residue is a concept from number theory, particularly in the study of modular arithmetic.
Quadratic residue codes are a class of error-correcting codes that are derived from the properties of quadratic residues in number theory. These codes are particularly notable in the field of coding theory for their efficiency and ability to detect and correct errors in transmitted data. ### Definition A quadratic residue modulo a prime \( p \) is an integer \( a \) such that there exists some integer \( x \) satisfying the equation \( x^2 \equiv a \mod p \).
Zolotarev's lemma is a result in number theory, particularly in the area of the distribution of prime numbers. It is often used in the context of modular forms and the study of certain types of power sums. The lemma is named after the Russian mathematician V. G. Zolotarev.
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 **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 \( n \).
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 **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 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 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 \( n \).
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.
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.
Vantieghem's theorem is not a widely recognized theorem in mathematics or science, and it seems that there may be some confusion regarding the name. It's possible that it's a misspelling or miscommunication of a different theorem or concept. If you're referring to a specific area of mathematics or a particular field (such as graph theory, number theory, etc.
The Vedic square is a mathematical construct that is derived from ancient Indian mathematics, specifically from the Vedic texts. It is essentially a multiplication table that showcases the results of multiplying numbers from 1 to 9, but it is unique in its arrangement and the patterns it reveals. To create a Vedic square, you typically follow these steps: 1. **Construct a 9x9 grid** where both the rows and columns represent the numbers 1 through 9.
