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.

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.).

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.

The term "Ultraparallel theorem" is not widely recognized in established mathematical literature or common mathematical terminology. However, it is possible that you are referring to a theorem related to non-Euclidean geometries or the properties of parallel lines. In the context of hyperbolic geometry, for example, two lines may be defined as "ultraparallel" if they do not intersect and are not parallel in the sense used in Euclidean geometry.

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 term "Theorem of the cube" is not widely recognized in mathematics as a specific theorem. However, it could refer to various concepts depending on the context.

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.

Soddy's hexlet is a configuration in geometry involving three circles that are tangent to each other in a specific way. Named after the British chemist Frederick Soddy, who explored this arrangement in connection with the theory of circles, Soddy's hexlet refers to the construction of two smaller circles that are tangent to three larger circles, along with two additional larger circles that touch the three originals.

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 Riemannian Penrose inequality is a result in differential geometry and general relativity that relates the total mass of a Riemannian manifold with boundary to the area of its boundary. It is an extension of the classical Penrose inequality, which is a key result in the theory of general relativity regarding the mass of gravitational systems.

The Petersen–Morley theorem is a result in graph theory that concerns the structure of certain types of graphs. It states that for every sufficiently large graph, if it contains no complete subgraph \( K_n \) of size \( n \), then the graph can be colored with \( n-1 \) colors such that no two adjacent vertices share the same color. The theorem is particularly relevant when discussing the properties of planar graphs and colorability.