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.
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.
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 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.
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 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.
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 \( p \).
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.
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.
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.
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.