Number theory stubs

In the context of Wikipedia and other collaborative online encyclopedias, a "stub" is a type of article that is considered incomplete or lacking in detail. A "Number theory stub" specifically refers to a very brief article related to the field of number theory—a branch of pure mathematics devoted to the study of the integers and their properties. Stubs typically provide only basic information or a limited overview of the topic, and they are often marked with a template indicating that they need expansion.

The Adleman–Pomerance–Rumely (APR) primality test is a deterministic algorithm for determining whether a given number is prime. It was developed by Leonard Adleman, Carl Pomerance, and Michael Rumely and is notable for its efficiency and robust theoretical foundation.

The Ankeny–Artin–Chowla congruence is a result in number theory concerning prime numbers and their distributions. Specifically, it deals with the congruence relationship of prime numbers in the context of quadratic residues. The conjecture can be stated as follows: For any odd prime \( p \) and any integer \( a \) that is relatively prime to \( p \), there exists a prime \( q \equiv a \pmod{p} \).

Arithmetic varieties, in the context of algebraic geometry, refer to varieties defined over number fields or more general arithmetic fields, and they can be studied using both algebraic techniques and number theoretic methods. These varieties are often associated with Diophantine equations, which seek integer or rational solutions to polynomial equations. More formally, an arithmetic variety is an algebraic variety defined over the field of rational numbers \( \mathbb{Q} \) or over more general number fields.

Bauerian extension is a concept in the field of functional analysis and related areas of mathematics, particularly concerning the extension of linear operators or functionals from one space to another. It is named after the mathematician Friedrich Bauer. In a more specific context, consider the problem of extending bounded linear functionals defined on a subspace of a Banach space to the entire space.

Bonse's inequality is a mathematical result related to number theory, particularly in the context of prime numbers. Specifically, it provides a bound on the distribution of prime numbers and related sequences. The inequality asserts that for every positive integer \( n \), the sum of the reciprocals of the prime numbers up to \( n \) diverges logarithmically.

The Brauer–Siegel theorem is a result in number theory concerning the behavior of the class numbers of number fields and their corresponding zeta functions. It specifically addresses the relationship between the growth of the class numbers of number fields and their degrees of extensions over the rational numbers.

Brocard's conjecture is a hypothesis in number theory proposed by the mathematician Henri Brocard in 1876. It suggests that there are only a finite number of natural numbers \( n \) such that the expression \( n! + 1 \) (the factorial of \( n \) plus one) is a perfect square. In mathematical terms, Brocard's conjecture can be stated as: There are only finitely many integers \( n \) such that: \[ n!

The term "broken diagonal" can refer to different concepts depending on the context. Here are a few possible meanings: 1. **Mathematics or Geometry**: In geometry, a broken diagonal may refer to a piecewise linear path in a grid or a geometric figure that consists of segments forming a diagonal-like shape but is not a straight line. For instance, in a geometric grid, a broken diagonal could zigzag from one corner of a rectangle to the opposite corner.

The Brumer bound is a concept from number theory, specifically in the study of algebraic number fields and their units. In the context of Class Field Theory and the study of the structure of units in the ring of integers of a number field, the Brumer bound provides a way to estimate the size of units in the ring of integers of an algebraic number field. More formally, it is a bound on the regulator of the unit group of the field's integers.

A circular prime is a particular type of prime number that remains prime when its digits are rotated in all possible ways. For example, let's consider the prime number 197. Its digit rotations are 197, 971, and 719, and since all of these numbers are prime, 197 is classified as a circular prime. To give another example, the number 13 is a circular prime because its rotations (13 and 31) are both prime numbers.

Continued fraction factorization refers to a mathematical method that expresses a number or a function as a continued fraction, especially in the context of factorizing algebraic expressions or certain types of numbers. Continued fractions are an alternative way to represent real numbers or rational numbers in the form of an infinite sequence of fractions.

Cyclotomic units are a special class of elements in the field of algebraic number theory, particularly within the context of cyclotomic fields. Cyclotomic fields are extensions of the rational numbers obtained by adjoining a primitive \( n \)-th root of unity, denoted as \( \zeta_n \), to the rationals \( \mathbb{Q} \).

The "Diamond Operator" often refers to two different concepts in programming and computer science, depending on the context: 1. **In Java Generics**: The Diamond Operator (`<>`) was introduced in Java 7 to simplify the use of generics.

The Dudley triangle, also known as the Dudley area or Dudley triangle concept, refers to a geographic and demographic model that describes three areas of interconnected significance in a particular region. This term is often used in discussions about urban planning, economic development, and social demographics. In some contexts, particularly in the UK, the Dudley triangle may refer to a specific area within the town of Dudley, located in the West Midlands, encompassing various neighborhoods or districts.

The term "Eigencurve" is not widely recognized and may refer to specific concepts or terminologies in various scientific or mathematical contexts. However, it's possible that it pertains to topics like eigenvalues/eigenvectors in linear algebra or certain applications in data science, machine learning, or computer vision, where curves or functions are analyzed through eigendecomposition techniques.

An elementary number refers to a number that is part of basic arithmetic and number theory—specifically, it typically refers to the integers, rational numbers, or certain simple constructs in mathematics. The term can also refer to specific types of numbers, such as the natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, ...), negative integers, or even certain classes of rational numbers.

Faltings' product theorem is a significant result in the field of arithmetic geometry, particularly concerning the theory of abelian varieties. It is a part of Faltings' broader work on the arithmetic of abelian varieties and their relation to rational points and Galois representations. In essence, Faltings' product theorem deals with the structure of the product of abelian varieties over a number field.

"Fermat's Last Theorem" is a book written by Simon Singh, published in 1997. The book explores the history and significance of Fermat's Last Theorem, which asserts that there are no three positive integers \(a\), \(b\), and \(c\) that satisfy the equation \(a^n + b^n = c^n\) for any integer value of \(n\) greater than 2.

Fricke involution is a concept found in the context of modular forms and algebraic geometry, particularly in relation to the study of modular curves. It is a specific type of involution—meaning it is an operation that can be applied twice to return to the original state—defined on the upper half-plane or on modular forms.

A Gaussian rational, also known as a Gaussian integer, is a complex number of the form \( a + bi \), where \( a \) and \( b \) are both rational numbers (i.e., they can be expressed as a fraction of integers), and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

The term "genus character" typically refers to the distinguishing features or characteristics that define a genus in biological classification. In taxonomy, the genus is a rank in the hierarchical classification system that groups species that are closely related to each other. Genus characters can include a variety of traits such as: 1. **Morphological Features:** These are physical characteristics, such as size, shape, structure, and color of the organisms that belong to that genus.

In mathematics, particularly in algebraic geometry and number theory, the term "genus field" typically refers to a field related to the concept of genus in the context of algebraic curves. 1. **Genus of a Curve**: The genus of an algebraic curve is a topological invariant that represents roughly the number of "holes" in the curve. For instance, a sphere has genus 0, while a torus has genus 1.

The Graß conjecture, also known as the Graß problem, is a problem in number theory related to prime numbers. Specifically, it posits a certain property of the primes in relation to their distribution. The conjecture asserts that for any integer \( n \), there exist infinitely many primes that can be expressed in the form \( n^2 + k \), for \( k \) being a positive integer that is not a perfect square.

A Gregory number refers to a specific type of number that is related to the Gregory series or Gregory-Leibniz series, which is an infinite series that can be used to estimate the value of π (pi).

The "Hexagonal Tortoise Problem" is a common conceptual or computational exercise often found in recreational mathematics or programming challenges. It involves a tortoise that moves on a hexagonal grid, typically starting from a specific point and moving in various directions based on certain rules. The problem usually requires finding a path, counting the number of distinct cells visited, or calculating possible movements. In a more specific context, the problem may involve defining how the tortoise moves (e.g.

The Igusa variety is a construct in the context of algebraic geometry and number theory, specifically related to the theory of abelian varieties and modular forms. It arises in the study of certain geometric objects associated with modular forms, where the Igusa variety serves as a parameter space for certain types of abelian varieties.

Ihara's lemma is a result in number theory, specifically in the study of zeta functions of curves over finite fields. The lemma relates the zeta function of a geometrically irreducible smooth projective curve over a finite field to the properties of its function field.

The Katz–Lang finiteness theorem is a result in algebraic geometry, specifically in the area of algebraic stacks and their cohomology. It provides conditions under which the set of isomorphism classes of certain algebraic objects can be shown to be finite. The theorem primarily concerns the situation involving stable maps to a projective variety (often referred to as the target variety), and it is particularly important in the context of counting curves.

The Koecher–Maass series is a mathematical series that arises in the context of the theory of modular forms and automorphic forms. It is named after mathematicians Martin Koecher and Hans Maass, who contributed to the understanding of modular forms and their properties. The series itself is typically associated with the theory of modular forms on the upper half-plane. These forms are complex functions that are not only continuous but also satisfy certain transformation properties under the action of the modular group.

Legendre's conjecture is an unsolved problem in number theory that concerns the distribution of prime numbers. It posits that there is at least one prime number between every pair of consecutive perfect squares.

The Local Trace Formula is a significant result in the fields of number theory and representation theory, particularly in the study of automorphic forms and L-functions. It relates the trace of an operator on a space of functions to geometric and number-theoretic data associated with a locally symmetric space. In more specific terms, the Local Trace Formula often appears in the context of the theory of L-functions and automorphic representations.

Lévy's constant, typically denoted as \( L \), is a mathematical constant that appears in the context of probability theory and stochastic processes, particularly concerning the law of the iterated logarithm for random walks and other related processes. More specifically, Lévy’s constant is related to the distribution of the supremum of a Brownian motion.

"Magic star" can refer to several different concepts, depending on the context. Here are a few possible interpretations: 1. **Mathematics**: In number theory, a magic star is similar to a magic square. It consists of points arranged in a star shape where the sums of the numbers in a specific formation (such as lines or diagonals) all yield the same total.

Maillet's determinant is a concept from the field of differential geometry, specifically dealing with the properties of surfaces and curves in three-dimensional space. It is commonly associated with the study of the curvature of surfaces and how these surfaces can be represented in a mathematical framework. In explicit terms, Maillet's determinant is often referred to in the context of calculating the curvature or torsion of curves or surfaces defined parametrically.

The Manin obstruction is a concept in algebraic geometry and number theory, particularly in the context of rational points on algebraic varieties. It is named after Yuri Manin, who introduced it in the 1970s.

The Manin–Drinfeld theorem is a significant result in the field of algebraic geometry and number theory, particularly in the study of rational points on algebraic curves. It was developed independently by mathematicians Yuri Manin and Vladimir Drinfeld in the 1970s. The theorem deals with the existence and structure of rational points on certain types of algebraic varieties, especially in the context of curves defined over non-closed fields, such as the rational numbers \(\mathbb{Q}\).

Mazur's control theorem is a result in the field of dynamical systems that deals with the stabilization of nonlinear systems. The theorem is often associated with the control of nonlinear systems using feedback mechanisms, particularly in the context of providing a way to ensure that the system's trajectories converge to a desired equilibrium point or set. More specifically, Mazur's theorem can be summarized as follows: 1. **Setup**: Consider a nonlinear dynamical system that exhibits some form of chaotic or complex behavior.

The Mestre bound is an important concept in the field of algebraic geometry, particularly in the study of rational points on algebraic varieties. It specifically relates to the distribution of rational points on certain types of varieties defined over number fields. More formally, the Mestre bound gives estimates on the number of rational points of bounded height on a projective algebraic variety. The bound can be particularly useful when analyzing the rational points on curves, notably elliptic curves and abelian varieties.

In recreational mathematics, a **minimal prime** refers to a prime number that has certain minimal properties, often in the context of a specific mathematical structure or problem. While the term "minimal prime" may not have a universally agreed-upon definition, one common interpretation is that it may describe the smallest prime number in a particular set or sequence that meets specific criteria. For example, in the context of prime numbers, the smallest prime (which is 2) could be referred to as a minimal prime.

The Miyawaki method, named after Japanese botanist Akira Miyawaki, is a technique for creating dense, native forests in a short amount of time. While "Miyawaki lift" may not be a standard term, it’s possible that it refers to the benefits or effects of applying the Miyawaki method to urban or degraded landscapes, leading to improved biodiversity, ecosystem restoration, and carbon sequestration.

A modular unit generally refers to a standardized and interchangeable component or system that can be combined with other modular units to form a larger, more complex structure or functioning system. This concept is applied across various fields, including architecture, manufacturing, software development, and education.

A monogenic field is a concept that arises in the context of algebraic number theory and field theory. The term generally refers to a field extension that is generated by a single element, also known as a primitive element.

The term "Multimagic cube" typically refers to a type of mathematical puzzle that extends the concept of a magic square or magic cube into higher dimensions. A magic cube is a three-dimensional arrangement of numbers in which the sums of the numbers in each row, column, and diagonal (in all three dimensions) are equal to a constant known as the magic constant.

In number theory, the **normal order** of an arithmetic function describes the typical or average asymptotic behavior of the function across integers. More formally, an arithmetic function \( f(n) \) is said to have a normal order \( g(n) \) if, for almost all integers \( n \), \( f(n) \) is approximately equal to \( g(n) \) in a certain sense.

Octic reciprocity is a concept in number theory, particularly in the field of algebraic number theory, which extends the idea of reciprocity laws for quadratic residues (the classical quadratic reciprocity) to higher powers. While the classic quadratic reciprocity law, proven by Carl Friedrich Gauss, deals with the solvability of certain congruences involving squares (i.e., second powers), octic reciprocity focuses on eighth powers.

Overconvergent modular forms are a special class of modular forms that arise in the context of p-adic analysis and arithmetic geometry, particularly in relation to the theory of p-adic modular forms and overconvergent systems of forms. In classical terms, a modular form is a complex analytic function on the upper half-plane that satisfies specific transformation properties under the action of a congruence subgroup of \( SL(2, \mathbb{Z}) \).

The Parshin chain is a concept in the realm of algebraic geometry and is named after the mathematician A. N. Parshin. It is related to the study of algebraic curves and surfaces, particularly in terms of their properties and invariants as they relate to various algebraic fields and number fields. In a more specific context, the Parshin chain can refer to particular types of sequences of algebraic objects that exhibit certain coherency and structural properties.

Porter's constant, also known as Porter's constant of diffusion, describes the rate of diffusion of small molecules through a particular medium. In the context of scientific studies, it is often associated with the diffusion of gases or solutes in liquids or solids. The concept can be linked to the general principles of diffusion, which are encapsulated in Fick's laws.

The concept of rational reciprocity is often discussed in the context of mathematical fields like number theory, particularly in relation to the reciprocity laws concerning quadratic residues and extensions to higher-degree polynomial equations. The classical form of the reciprocity law is known as **quadratic reciprocity**, which states relationships between the solvability of two quadratic equations in a modular arithmetic setting.

Raynaud's isogeny theorem is an important result in the field of algebraic geometry, particularly in the study of abelian varieties and their isogenies. The theorem establishes a connection between abelian varieties, specifically abelian varieties that are defined over a number field or a finite field, and their isogenies, which are morphisms between these varieties that preserve their group structure and have finite kernel.

The Second Hardy–Littlewood conjecture, also known as the "2-ary Goldbach conjecture," is an unsolved problem in number theory that is concerned with the representation of even integers as sums of prime numbers. Specifically, it builds upon the ideas found in the original Goldbach conjecture. The conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

Shimura's reciprocity law is a profound result in the theory of numbers, particularly in the context of modular forms and the Langlands program. It generalizes classical reciprocity laws, such as those established by Gauss and later by Artin, to a broader setting involving Shimura varieties and abelian varieties. In essence, Shimura’s reciprocity law connects the arithmetic properties of abelian varieties defined over number fields to the values of certain automorphic forms.

A Shimura subgroup is a certain type of subgroup that arises in the context of Shimura varieties, which are higher-dimensional generalizations of modular curves. Shimura varieties play an important role in number theory and have connections to arithmetic geometry, automorphic forms, and the Langlands program.

The Siegel G-function is a complex function that arises in the theory of analytic number theory, particularly in the area of modular forms and automorphic forms. It is named after the German mathematician Carl Ludwig Siegel, who made significant contributions to number theory and related fields. In a broad sense, the Siegel G-function can be viewed as a generalization of the classical gamma function and is associated with several variables.

The Siegel-Weil formula is a significant result in the realm of number theory and the theory of automorphic forms. It relates to the theory of modular forms and L-functions and provides a bridge between number theory, algebraic geometry, and representation theory. The essence of the Siegel-Weil formula lies in establishing a deep connection between certain arithmetic objects (like algebraic cycles) and special values of L-functions associated with these objects.

A **super-prime** is a special type of prime number that is itself prime and also has a prime index in the ordered sequence of all prime numbers.

Szpiró's conjecture is a hypothesis in number theory regarding the distribution of prime numbers in relation to certain algebraic curves, specifically those defined over number fields. Formulated by the mathematician Szpiro in the context of elliptic curves, it establishes a connection between the height of a point on the curve and the number of rational points of bounded height.

The Torsion Conjecture is related to algebraic geometry and the theory of elliptic curves, particularly in the context of the arithmetic and geometric properties of algebraic varieties. Specifically, it concerns the relationship between the torsion points of an elliptic curve defined over the rational numbers and the set of rational points on the curve. The conjecture posits that the torsion subgroup of an elliptic curve over the rational numbers has a structure constrained by the properties of the curve itself.

A totally imaginary number field is a specific type of number field where every element of the field has its conjugates (in terms of field embeddings into the complex numbers) lying on the imaginary axis. More precisely, a number field is a finite extension of the field of rational numbers \(\mathbb{Q}\).

An undulating number is a number where the digits alternately increase and decrease. More formally, a number is considered undulating if, for every pair of adjacent digits in the number, either the left digit is greater than the right digit or the left digit is less than the right digit, with no two adjacent digits being equal.

In mathematics, particularly in linear algebra and functional analysis, the term "unit function" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Unit Function in Physics and Engineering**: In the context of signals, the "unit function" often refers to the unit step function (Heaviside function), which takes the value of 0 for negative inputs and 1 for non-negative inputs.

Waring's prime number conjecture is an extension of Waring's problem, which originally deals with the representation of natural numbers as sums of a fixed number of powers of natural numbers. Specifically, Waring's problem states that for any natural number \( k \), there exists a minimum integer \( g(k) \) such that every natural number can be expressed as the sum of at most \( g(k) \) \( k \)-th powers of natural numbers.