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.

Lindström's theorem is a significant result in model theory, a branch of mathematical logic that deals with the relationships between formal languages and their interpretations, or models. Formulated by Per Lindström in the 1960s, the theorem characterizes the logical systems that enjoy certain completeness and categoricity properties, specifically those known as the "Lindström properties.

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.

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.

Robinson's joint consistency theorem is a result in the field of decision theory and economics related to the consistency of preferences and the representation of preferences by a utility function. The theorem addresses the question of how to represent preferences over a set of choices that may vary according to certain parameters. Specifically, it deals with the conditions under which a joint distribution of choices can be consistent with the preferences of agents when making those choices.

Frege's theorem is a significant result in the foundations of mathematics and logic, attributed to the German mathematician and philosopher Gottlob Frege. It establishes the connection between logic and mathematics, specifically concerning the foundations of arithmetic. At its core, Frege's theorem asserts that the basic propositions of arithmetic can be derived from purely logical axioms and definitions. More specifically, it shows that the arithmetic of natural numbers can be defined in terms of logic through the formalization of the concept of number.

The Deduction Theorem is a fundamental principle in propositional logic and mathematical logic. It establishes a relationship between syntactic proofs and semantic entailment. The theorem can be stated as follows: If a formula \( B \) can be derived from a set of premises \( \Gamma \) along with an additional assumption \( A \), then it is possible to infer that the implication \( A \rightarrow B \) can be derived from the premises \( \Gamma \) alone.

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.

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.

The Thom conjecture, proposed by mathematician René Thom in the 1950s, relates to topology and singularity theory. Specifically, it concerns the structure of non-singular mappings between manifolds and the conditions under which certain types of singularities can occur. The conjecture asserts that every real-valued function defined on a manifold can be approximated by a function that has a certain type of "generic" singularity.

The Spherical Law of Cosines is a fundamental theorem in spherical geometry, which deals with the relationships between the angles and sides of spherical triangles (triangles drawn on the surface of a sphere). Specifically, it is used to relate the lengths of the sides of a spherical triangle and the cosine of one of its angles.

The Skoda–El Mir theorem is a result in complex analysis, specifically in the theory of several complex variables and the study of holomorphic functions. It pertains to the properties of holomorphic functions defined on complex manifolds, particularly focusing on the behavior of such functions near their zero sets. In essence, the theorem addresses the relationships between the zero sets of holomorphic functions and their implications for the analyticity and continuity of these functions.

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.

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 Lickorish–Wallace theorem is a result in the field of topology, specifically in the study of 3-manifolds. This theorem provides a criterion for when a connected sum of 3-manifolds can be represented as a connected sum of prime 3-manifolds.

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.

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 \( p-1 \).

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.