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Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation $a^n + b^n = c^n$ for any integer value of n greater than 2. The note by Pierre de Fermat in 1637 stating he had a proof without enough margin to write it led to over 350 years of mystery until Andrew Wiles provided a proof in 1995 using elliptic curves and modular forms. This theorem has a profound influence on the development of number theory.

The Happy Ending Problem, proposed by Esther Klein in 1933, involves finding the smallest number of points in the plane (in general position) that guarantees the existence of a convex quadrilateral. The problem introduced important concepts in combinatorial geometry and led to the formulation of Ramsey Theory.

The Twin Prime Conjecture suggests that there are infinitely many prime numbers p such that p+2 is also prime. The concept of twin primes was first recorded by Alphonse de Polignac in 1849, and it has been a topic of significant research in number theory. Despite substantial numerical evidence and partial results, the general case remains unproven.

The Lonely Runner Conjecture states that given k runners on a track of unit length, moving at different constant speeds, there will be a time when each runner is at least a distance 1/k from every other runner. Initially posed in the context of diophantine approximations, this problem has implications in number theory and combinatorics.

The Angel Problem is a game played on an infinite chessboard, where the goal is to determine whether the angel, which can make large leaps, can always evade the devil, who aims to block its path. This problem, proposed by John Horton Conway in 1982, has been solved, illustrating computation techniques in combinatorial game theory.

The Erdős Distinct Distances Problem asks for the minimum number of distinct distances between any set of n points in the plane. Formulated by Paul Erdős in 1946, this question depends on the structure of the point set and is core to discrete geometry and combinatorial geometry. While progress has been made, the exact answer for every n is still unknown.

The Moving Sofa Problem seeks to determine the shape with the largest area that can move around a right-angled corner in a corridor with unit width. First posed by Leo Moser in 1966, this problem remains unsolved and has applications in optimization and geometric analysis.

The Continuum Hypothesis is concerned with the possible sizes of infinite sets, specifically suggesting there's no set size between the infinity of the integers and the infinity of the real numbers. Proposed by Georg Cantor in 1878, it was the first problem on Hilbert's list of 23 unsolved problems, presented in 1900. The hypothesis has implications for set theory and was proved to be independent of the standard axiomatization of set theory (ZFC) by Paul Cohen in 1963.

The Birch and Swinnerton-Dyer Conjecture pertains to the set of rational solutions to equations defining an elliptic curve. It proposes a way to determine the existence of an infinite number of rational points on an elliptic curve based on the behavior of an associated function as it approaches the origin. This is one of the Millennium Prize Problems and is pivotal in the field of number theory, especially in the study of elliptic curves.

Hilbert's Third Problem questioned whether it's possible to decompose a tetrahedron into finitely many tetrahedra and reassemble them into a square. This problem was resolved in the negative by Max Dehn in 1900, identifying the Dehn invariant as an obstruction to such decompositions. It was one of the first major problems in the early development of geometric topology.

The Inscribed Square Problem, also known as Toeplitz' conjecture or squaring the circle, asks whether every closed curve without self-intersections contains all four vertices of some square. It remains unsolved, although it has been shown to be true for various types of curves. Its resolution would further our understanding of geometric properties and combinatorial aspects of curves.

The Goldbach Conjecture asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Proposed by Christian Goldbach in 1742, it remains one of the oldest unsolved problems in number theory and has been tested by computers for large even numbers but lacks a general proof.

The Navier-Stokes equations describe the motion of viscous fluid substances and are fundamental in fluid dynamics. The existence and smoothness problem asks whether solutions to the Navier-Stokes equations always exist and are smooth in three dimensions. This is another one of the Millennium Prize Problems for which a solution or disproof carries a $1 million reward.$

The Collatz Conjecture, also known as the 3n+1 problem, hypothesizes that the sequence defined by the iterative process: $n \rightarrow n/2$ if n is even, and $n \rightarrow 3n+1$ if n is odd, will eventually reach the number 1 for all positive integer inputs. Proposed by Lothar Collatz in 1937, the problem has been extensively tested, but no proof has yet been found, nor has a counterexample been produced.

The Hadwiger-Nelson Problem concerns the minimum number of colors required to color the plane such that no two points at distance exactly 1 from each other are the same color. Posed by Hugo Hadwiger and Edward Nelson in 1950, the number of required colors is known to be between 4 and 7, but the exact number remains undetermined. This problem is important in graph coloring and geometric graph theory.

The Langlands Program is a series of conjectures and theories relating Galois groups in algebraic number theory to automorphic forms and representations of algebraic groups over global fields. Initiated by Robert Langlands in the late 1960s, it aims to create a grand unified theory connecting different areas of mathematics and has been influential in both number theory and representation theory.

The Poincaré Conjecture was a central problem in topology, stating that a closed three-dimensional manifold that is homotopy equivalent to a sphere is also homeomorphic to a sphere. Proposed by Henri Poincaré in 1904, it was solved by Grigori Perelman in 2003 using Richard S. Hamilton's Ricci flow. Perelman's proof stands as a landmark result in the field of geometric topology.

The Beal Conjecture proposes that if $A^x + B^y = C^z$, where A, B, C, x, y, and z are positive integers and x, y, z > 2, then A, B, and C must have a common prime factor. Posed by Andrew Beal in 1993, a substantial monetary prize is offered to anyone providing a proof or counterexample, adding to the allure for mathematicians. The conjecture generalizes Fermat's Last Theorem and has yet to be proven.

The Kepler Conjecture is concerned with the arrangement of congruent spherical objects (like cannonballs) packed into a given space and posits that no arrangement is more efficient than a pyramid-like lattice. First stipulated by Johannes Kepler in 1611, the conjecture was proved by Thomas Hales in 1998 using a combination of mathematical and computational methods, a significant step in the study of packing problems and discrete geometry.

The Riemann Hypothesis conjectures that all non-trivial zeros of the Riemann zeta function have a real part of 1/2. Formulated by Bernhard Riemann in 1859, it's a central problem in number theory and has implications in the distribution of prime numbers. It remains unsolved and is one of the seven Millennium Prize Problems with a $1 million prize for a correct proof.$

In theoretical physics and mathematics, the Yang-Mills Existence and Mass Gap problem involves proving that for Yang-Mills fields in the vacuum state, there exists a mass gap over which the fields become trivial. This problem, one of the Millennium Prize Problems, has implications for quantum field theory and the standard model of particle physics.

The P=NP Problem is a major unsolved problem in computer science that deals with the relationship between the complexity classes P (problems solvable in polynomial time) and NP (problems verifiable in polynomial time). It has vast implications across various fields, including cryptography, algorithm design, and market equilibria. A solution to this problem would have profound consequences for mathematics, science, and technology.

The Banach-Tarski Paradox is a result in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-measurable pieces and reassembled into two solid balls identical to the original. Proposed by Stefan Banach and Alfred Tarski in 1924, this paradox highlights the counterintuitive consequences of the axiom of choice in set theory.

The Four Color Theorem posits that any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. First introduced in 1852 and proved in 1976 by Kenneth Appel and Wolfgang Haken, it was the first major theorem to be proved using a computer. It has significant implications in topology and graph theory.

Another name for the Collatz Conjecture; the Syracuse Problem involves an integer sequence defined by a recursive process similar to that of the Collatz sequence. Despite its simple formulation, this problem baffles mathematicians due to its seemingly chaotic behavior and remains unsolved.

The Monster Group, or Fischer-Griess Monster, is the largest sporadic simple group in group theory, containing over $8 \times 10^{53}$ elements. It was predicted by Bernd Fischer and Robert Griess in the 1970s and plays a central role in the classification of finite simple groups, with intriguing connections to string theory and theoretical physics.

The Hodge Conjecture is a major unsolved problem in algebraic geometry that deals with the relationship between algebraic cycles and the cohomology of a smooth projective algebraic variety. Posed by William Vallance Douglas Hodge in the 20th century, it suggests certain de Rham cohomology classes are algebraic. It is one of the Millennium Prize Problems.