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.

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.

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.

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.

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.

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 \( p-1 \).

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.

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.

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 \( n \).

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

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.

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.

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.

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).

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

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.

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.

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.

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.

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.