Theorems in number theory

In number theory, theorems are established propositions that are proven to be true based on previously accepted statements, such as axioms and previously proven theorems. Number theory itself is a branch of mathematics that deals with the properties and relationships of numbers, especially integers.

In number theory, a lemma is a proven statement or proposition that is used as a stepping stone to prove a larger theorem. The term "lemma" comes from the Greek word "lemma," which means "that which is taken" or "premise." Lemmas can be thought of as auxiliary results that help in the development of more complex arguments or proofs.

Euclid's lemma is a fundamental statement in number theory that relates to the properties of prime numbers and divisibility. It states: **If a prime number \( p \) divides the product of two integers \( a \) and \( b \) (i.e., \( p \mid (a \cdot b) \)), then \( p \) must divide at least one of those integers \( a \) or \( b \) (i.e.

The Lifting-the-exponent lemma (LTE) is a mathematical result in number theory that provides conditions under which the highest power of a prime \( p \) that divides certain expressions can be easily determined. It simplifies the computation of \( v_p(a^n - b^n) \) and related expressions, where \( v_p(x) \) denotes the p-adic valuation, which gives the exponent of the highest power of \( p \) that divides \( x \).

The "15 theorem" and "290 theorem" might refer to specific mathematical theorems or results, but the terminology you've used is not standard in mathematics. To help you better, I would need more context about what these theorems pertain to or which area of mathematics they relate to (e.g., number theory, geometry, algebra, etc.).

Baker's theorem pertains to the field of complex analysis, specifically dealing with functions that can be expressed through power series. More formally, it relates to the growth of meromorphic functions, which are functions that are holomorphic (complex differentiable) everywhere except for a set of isolated poles.

Behrend's theorem is a result in the field of combinatorial number theory, particularly concerning the distribution of numbers that are free of a specific type of arithmetic progression.

Carmichael's theorem, also known as Carmichael's function, deals with properties of groups and relates to the structure of finite abelian groups. Specifically, it provides a way to determine the order of elements in a group.

The Davenport–Erdős theorem is a result in additive number theory, specifically concerning the sum sets of subsets of integers. It states that if \( A \) is a subset of the natural numbers \( \mathbb{N} \) with finite positive upper density, then the set of all finite sums of elements of \( A \) (i.e.

The Davenport-Schmidt theorem is a result in number theory that deals with the distribution of integers that can be expressed as the sum of two squares. Specifically, the theorem states that for any positive integer \( n \) that is not of the form \( 4^k(8m + 7) \) for nonnegative integers \( k \) and \( m \), there are infinitely many integers that can be represented as a sum of two squares.

Dirichlet's approximation theorem is a result in number theory that provides a way to find rational approximations to real numbers.

Eisenstein's theorem, often referred to in the context of mathematics and number theory, primarily concerns the factorization of polynomials with integer coefficients. It provides a criterion for determining whether a polynomial is irreducible over the field of rational numbers (or, equivalently, over the integers).

The Euclid–Euler theorem, also known as Euler's theorem in the context of number theory, relates to the area of geometry and can be specifically described in two ways.

Faltings's theorem, proven by Gerd Faltings in 1983, is a significant result in number theory and algebraic geometry. The theorem states that: **For a given algebraic curve defined over the rationals (or more generally, over any number field), there are only finitely many rational points on the curve, provided the genus of the curve is greater than or equal to 2.

Fermat's Last Theorem states that there are no three positive integers \(a\), \(b\), and \(c\) that can satisfy the equation \(a^n + b^n = c^n\) for any integer value of \(n\) greater than 2.

"Fermat's Last Tango" is a musical by Tara O'Brady (music and lyrics) and Jay O'Brady (book) that premiered in 2000.

Fermat's Last Theorem, which states 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, has inspired various works of fiction that explore themes of mathematics, obsession, and the human condition.

Fermat's Last Theorem states that there are no integers \( x, y, z \) and \( n \) greater than 2 such that \[ x^n + y^n = z^n. \] Fermat famously wrote in the margin of his copy of Diophantus' *Arithmetica* that he had discovered "a truly marvelous proof" of this theorem, which he claimed was too large to fit in the margin.

Fermat's Last Theorem states 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. The theorem was famously conjectured by Pierre de Fermat in 1637 and was not proven until Andrew Wiles completed his proof in 1994.

Andrew Wiles's proof of Fermat's Last Theorem, completed in 1994, is a profound development in number theory that connects various fields of mathematics, particularly modular forms and elliptic curves. Fermat's Last Theorem states that there are no three positive integers \(a\), \(b\), and \(c\) such that \(a^n + b^n = c^n\) for any integer \(n > 2\).

The Gelfond–Schneider theorem is a fundamental result in transcendental number theory, established by Aleksandr Gelfond and Richard Schneider in the 1930s.

Hurwitz's theorem in number theory, specifically concerning the distribution of integers with respect to their divisibility properties, states that for any integer \( n \), the number of representations of \( n \) as a sum of two positive squares, denoted \( r_2(n) \), can be expressed in terms of the prime factorization of \( n \).

Kaplansky's theorem on quadratic forms is a significant result in the theory of quadratic forms over rings, particularly concerning the values that can be obtained by quadratic forms over certain fields. The theorem specifically states conditions under which a quadratic form can be represented as the sum of squares of linear forms. In particular, one of the most notable facets of Kaplansky's work on quadratic forms relates to the representation of forms over the integers and over various fields.

Kummer's theorem is a result in number theory that deals with the generating function of a specific type of polynomial, known as Kummer polynomials, and is related to the combinatorial interpretation of binomial coefficients and hypergeometric functions. The theorem broadly states conditions under which certain series can be expressed in terms of known functions or simpler forms.

Loch's theorem, in the context of mathematics, particularly in number theory, provides a result concerning the divisibility of certain numbers by others. Specifically, it states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then the order of \( a \) modulo \( p \) divides \( p-1 \).

Mahler's compactness theorem is a result in the field of mathematical logic, specifically in model theory. The theorem deals with the idea of compactness in relation to sets of first-order sentences. It essentially states that if every finite subset of a set of first-order sentences is satisfiable (i.e., there exists a model that satisfies all the sentences in that finite subset), then the entire set of sentences is also satisfiable.

Meyer's theorem is a result in the field of stochastic calculus, particularly dealing with semimartingales and their properties in the context of stochastic integration and Itô calculus. Specifically, the theorem provides conditions under which a process is a semimartingale and gives criteria for the convergence of stochastic integrals. In more detail, Meyer's theorem deals with certain types of stochastic processes, often focusing on the convergence of integrals involving local martingales.

The Modularity Theorem, which is a significant result in number theory, asserts a deep connection between elliptic curves and modular forms. Specifically, it states that every rational elliptic curve over the field of rational numbers is modular.

The Nagell–Lutz theorem is a result in the theory of Diophantine equations, specifically concerning the representation of integers as sums of powers of natural numbers. It states that if a prime \( p \) can be expressed as a sum of two square numbers, i.e.

Proizvolov's identity is a mathematical result related to combinatorics and, more specifically, to enumerative geometry and the study of plane partitions. It is named after the Russian mathematician Vyacheslav Proizvolov. In essence, Proizvolov's identity connects the counting of certain combinatorial structures, often through a generating function or through some algebraic identity. The identity can be used to derive results about integer partitions, multinomial coefficients, and more.

Ramanujan's congruences refer to a set of remarkable congruences related to partition numbers, which count the number of ways a given positive integer can be expressed as the sum of positive integers, without regard to the order of the summands.

Ribet's theorem is a fundamental result in number theory related to the Taniyama-Shimura-Weil conjecture, which is a key element in the proof of Fermat's Last Theorem. The theorem, proved by Ken Ribet in 1986, establishes a crucial connection between elliptic curves and modular forms.

Romanov's theorem refers to a result in the field of mathematics, specifically in the area of functional analysis or approximation theory. However, there may be various references and contexts in which "Romanov's theorem" is used, as the names of theorems can often relate to the work of specific mathematicians. One possible reference is the theorem related to the approximation of certain types of functions, often concerning the properties of interpolation or approximation in normed spaces.

Roth's theorem, established by mathematician Klaus Roth in 1951, is a significant result in the field of number theory, particularly in the study of arithmetic progressions and additive combinatorics. The theorem specifically deals with the distribution of rational approximations to irrational numbers. In its classical form, Roth's theorem states that if \(\alpha\) is an irrational number, then it cannot be well-approximated by rational numbers in a very precise way.

Serre's modularity conjecture, proposed by Jean-Pierre Serre in the 1980s, is a deep and influential hypothesis in the field of number theory, particularly concerning the relationship between modular forms and elliptic curves.

The Six Exponentials Theorem is a result in complex analysis and differential equations that deals with the solutions of certain classes of linear differential equations. It establishes conditions under which specific linear combinations of exponential functions can represent the solutions to these equations.

The Skolem–Mahler–Lech theorem is a result in number theory and in the study of sequences which concerns the behavior of integer sequences defined by linear recurrence relations. More specifically, it deals with the properties of the zeros of such sequences.

Sophie Germain's theorem is a result in number theory concerning prime numbers. It states that if \( p \) is a prime number, and \( 2p + 1 \) is also prime, then \( p \) is called a Sophie Germain prime, and \( 2p + 1 \) is called a safe prime.

The Subspace Theorem is a significant result in Diophantine approximation and algebraic geometry, primarily associated with the work of mathematician W. Michael M. Schmidt. It provides a strong criterion for understanding when certain types of linear forms in algebraic numbers can approximate other algebraic numbers closely.

The Turán–Kubilius inequality is a result in number theory and probabilistic number theory, often related to the distribution of prime numbers. It provides a bound on the probability that certain events, often concerning the sums of random variables, will occur.

The Von Staudt–Clausen theorem is a result in the field of number theory, particularly concerning the theory of continued fractions and the approximation of numbers. The theorem provides a way to express a specific class of numbers, notably the values of certain mathematical constants, as a sum involving continued fractions.

Zeckendorf's theorem states that every positive integer can be uniquely represented as a sum of one or more distinct non-consecutive Fibonacci numbers.