sâmbătă, 21 decembrie 2019

Number theory and the enchanted hexagon.







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  1. Teleman Dan dThelema29 decembrie 2019 la 15:13

    The shepherd's theorem.
      The number of sheep in a sheepfold remains, always the same, even if the shepherd does not know how to count.
      The number of sheep changes after the wolf visit.
    Le théorème du berger.
      Le nombre de moutons dans un mouton reste, toujours le même, même si le berger ne sait pas compter.
      Le nombre de moutons change après la visite du loup.
    Der Satz des Hirten.
      Die Anzahl der Schafe in einem Schafstall bleibt immer gleich, auch wenn der Hirte nicht zu zählen weiß.
      Die Anzahl der Schafe ändert sich nach dem Wolfsbesuch.
    Il teorema del pastore.
      Il numero di pecore in una pecora rimane, sempre lo stesso, anche se il pastore non sa come contare.
      Il numero di pecore cambia dopo la visita del lupo.
    Теорема Пастуха.
      Количество овец в овчарне остается неизменным, даже если пастух не умеет считать.
      Количество овец меняется после посещения волка.

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  2. Teleman Dan dThelema29 decembrie 2019 la 15:20

    Prime numbers are fundamentally important in mathematics. Watch this talk by Dr Vicky Neale (Mathematical Institute, University of Oxford) to discover some of the beautiful properties of prime numbers, and learn about some of the unsolved problems that mathematicians are working on today.

    This talk was given to an audience of 16-17 year olds.
    Oh! what a mediocre child's opinion.

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  3. Teleman Dan dThelema29 decembrie 2019 la 15:25

    Acest comentariu a fost eliminat de autor.

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  4. Teleman Dan dThelema29 decembrie 2019 la 15:27

    Number theory
    MATHEMATICS
    WRITTEN BY: William Dunham
    See Article History
    Alternative Title: higher arithmetic
    ARTICLE CONTENTS
    Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural of mathematical pursuits.


    Number theory
    QUICK FACTS
    KEY PEOPLE
    Carl Friedrich Gauss
    Pierre de Fermat
    Diophantus
    Paul Erdős
    Leonhard Euler
    Eudoxus of Cnidus
    Fibonacci
    David Hilbert
    Richard Dedekind
    Joseph-Louis Lagrange, comte de l'Empire
    RELATED TOPICS
    Mathematics
    Riemann hypothesis
    Twin prime conjecture
    Prime number theorem
    Fermat's last theorem
    Diophantine equation
    Fermat's theorem
    Waring's problem
    Lagrange's four-square theorem
    Birch and Swinnerton-Dyer conjecture
    Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background.

    Until the mid-20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. The advent of digital computers and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes, testing conjectures, and solving numerical problems once considered out of reach.
    https://www.britannica.com/science/number-theory

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  5. Teleman Dan dThelema29 decembrie 2019 la 15:44

    W

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  6. Teleman Dan dThelema29 decembrie 2019 la 15:45

    https://www.storyofmathematics.com/mathematicians.html

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  7. Teleman Dan dThelema29 decembrie 2019 la 15:47

    African First notched tally bones
    3100 BCE Sumerian Earliest documented counting and measuring system
    2700 BCE Egyptian Earliest fully-developed base 10 number system in use
    2600 BCE Sumerian Multiplication tables, geometrical exercises and division problems
    2000-1800 BCE Egyptian Earliest papyri showing numeration system and basic arithmetic
    1800-1600 BCE Babylonian Clay tablets dealing with fractions, algebra and equations
    1650 BCE Egyptian Rhind Papyrus (instruction manual in arithmetic, geometry, unit fractions, etc)
    1200 BCE Chinese First decimal numeration system with place value concept

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  8. Teleman Dan dThelema29 decembrie 2019 la 15:49

    ce value concept
    1200-900 BCE Indian Early Vedic mantras invoke powers of ten from a hundred all the way up to a trillion
    800-400 BCE Indian “Sulba Sutra” lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2
    650 BCE Chinese Lo Shu order three (3 x 3) “magic square” in which each row, column and diagonal sums to 15
    624-546 BCE Thales Greek Early developments in geometry, including work on similar and right triangles
    570-495 BCE Pythagoras Greek Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem
    500 BCE Hippasus Greek Discovered potential existence of irrational numbers while trying to calculate the value of √2

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  9. Teleman Dan dThelema29 decembrie 2019 la 15:51

    value of √2
    490-430 BCE Zeno of Elea Greek Describes a series of paradoxes concerning infinity and infinitesimals
    470-410 BCE Hippocrates of Chios Greek First systematic compilation of geometrical knowledge, Lune of Hippocrates
    460-370 BCE Democritus Greek Developments in geometry and fractions, volume of a cone
    428-348 BCE Plato Greek Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods
    410-355 BCE Eudoxus of Cnidus Greek Method for rigorously proving statements about areas and volumes by successive approximations
    384-322 BCE Aristotle Greek Development and standardization of logic (although not then considered part of mathematics) and deductive reasoning
    300 BCE Euclid Greek Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid’s Theorem on infinitude of primes
    287-212 BCE Archimedes Greek Formulas for areas of regular shapes, “method of exhaustion” for approximating areas and value of π, comparison of infinities
    276-195 BCE Eratosthenes Greek “Sieve of Eratosthenes”

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  10. Teleman Dan dThelema29 decembrie 2019 la 15:52

    comparison of infinities
    276-195 BCE Eratosthenes Greek “Sieve of Eratosthenes” method for identifying prime numbers
    262-190 BCE Apollonius of Perga Greek Work on geometry, especially on cones and conic sections (ellipse, parabola, hyperbola)
    200 BCE Chinese “Nine Chapters on the Mathematical Art”, including guide to how to solve equations using sophisticated matrix-based methods
    190-120 BCE Hipparchus Greek Develop first detailed trigonometry tables
    36 BCE Mayan Pre-classic Mayans developed the concept of zero by at least this time
    10-70 CE Heron (or Hero) of Alexandria Greek Heron’s Formula for finding the area of a triangle from its side lengths, Heron’s Method for iteratively computing a square root
    90-168 CE Ptolemy Greek/Egyptian Develop even more detailed trigonometry tables
    200 CE Sun Tzu Chinese First definitive statement of Chinese Remainder Theorem
    200 CE Indian Refined and perfected decimal place value number system
    200-284 CE Diophantus Greek Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns
    220-280 CE Liu Hui Chinese Solved linear equations

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  11. Teleman Dan dThelema29 decembrie 2019 la 15:53

    unknowns
    220-280 CE Liu Hui Chinese Solved linear equations using a matrices (similar to Gaussian elimination), leaving roots unevaluated, calculated value of π correct to five decimal places, early forms of integral and differential calculus
    400 CE Indian “Surya Siddhanta” contains roots of modern trigonometry, including first real use of sines, cosines, inverse sines, tangents and secants
    476-550 CE Aryabhata Indian Definitions of trigonometric functions, complete and accurate sine and versine tables, solutions to simultaneous quadratic equations, accurate approximation for π (and recognition that π is an irrational number)
    598-668 CE Brahmagupta Indian Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns
    600-680 CE Bhaskara I Indian First to write numbers in Hindu-Arabic decimal system with a circle for zero, remarkably accurate approximation of the sine function
    780-850 CE Muhammad Al-Khwarizmi Persian Advocacy of the Hindu numerals 1 - 9 and 0 in Islamic world, foundations of modern algebra, including algebraic methods of “reduction” and “balancing”, solution of polynomial equations up to second degree
    908-946 CE Ibrahim ibn Sinan Arabic Continued Archimedes

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  12. Teleman Dan dThelema29 decembrie 2019 la 15:54

    second degree
    908-946 CE Ibrahim ibn Sinan Arabic Continued Archimedes' investigations of areas and volumes, tangents to a circle
    953-1029 CE Muhammad Al-Karaji Persian First use of proof by mathematical induction, including to prove the binomial theorem
    966-1059 CE Ibn al-Haytham (Alhazen) Persian/Arabic Derived a formula for the sum of fourth powers using a readily generalizable method, “Alhazen's problem”, established beginnings of link between algebra and geometry
    1048-1131 Omar Khayyam Persian Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots, noted existence of different sorts of cubic equations
    1114-1185 Bhaskara II Indian Established that dividing by zero yields infinity, found solutions to quadratic, cubic and quartic equations (including negative and irrational solutions) and to second order Diophantine equations, introduced some preliminary concepts of calculus
    1170-1250 Leonardo of Pisa (Fibonacci) Italian Fibonacci Sequence of

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  13. Teleman Dan dThelema29 decembrie 2019 la 15:55

    calculus
    1170-1250 Leonardo of Pisa (Fibonacci) Italian Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)
    1201-1274 Nasir al-Din al-Tusi Persian Developed field of spherical trigonometry, formulated law of sines for plane triangles
    1202-1261 Qin Jiushao Chinese Solutions to quadratic, cubic and higher power equations using a method of repeated approximations
    1238-1298 Yang Hui Chinese Culmination of Chinese “magic” squares, circles and triangles, Yang Hui’s Triangle (earlier version of Pascal’s Triangle of binomial co-efficients)
    1267-1319 Kamal al-Din al-Farisi Persian Applied theory of conic sections to solve optical problems, explored amicable numbers, factorization and combinatorial methods
    1350-1425 Madhava Indian Use of infinite series of fractions to give an exact formula for π, sine formula and other trigonometric functions, important step towards development of calculus
    1323-1382 Nicole Oresme French System of rectangular

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  14. Teleman Dan dThelema29 decembrie 2019 la 15:56

    development of calculus
    1323-1382 Nicole Oresme French System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series
    1446-1517 Luca Pacioli Italian Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus
    1499-1557 Niccolò Fontana Tartaglia Italian Formula for solving all types of cubic equations, involving first real use of complex numbers (combinations of real and imaginary numbers), Tartaglia’s Triangle (earlier version of Pascal’s Triangle)
    1501-1576 Gerolamo Cardano Italian Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)
    1522-1565 Lodovico Ferrari Italian Devised formula for solution of quartic equations
    1550-1617 John Napier British Invention of natural logarithms, popularized the use of the decimal point, Napier’s Bones tool for lattice multiplication
    1588-1648 Marin Mersenne French Clearing house for mathematical thought during 17th Century, Mersenne primes (prime numbers that are one less than a power of 2)
    1591-1661 Girard Desargues French Early development of projective geometry and “point at infinity”, perspective theorem
    1596-1650 René Descartes French Development of

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  15. Teleman Dan dThelema29 decembrie 2019 la 15:57

    perspective theorem
    1596-1650 René Descartes French Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents
    1598-1647 Bonaventura Cavalieri Italian “Method of indivisibles” paved way for the later development of infinitesimal calculus
    1601-1665 Pierre de Fermat French Discovered many new numbers patterns and theorems (including Little Theorem, Two-Square Thereom and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory
    1616-1703 John Wallis British Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers
    1623-1662 Blaise Pascal French Pioneer (with Fermat) of probability theory, Pascal’s Triangle of binomial coefficients
    1643-1727 Isaac Newton British Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series
    1646-1716 Gottfried Leibniz German Independently developed infinitesimal calculus (his calculus notation is still used), also practical calculating machine using binary system (forerunner of the computer), solved linear equations using a matrix
    1654-1705 Jacob Bernoulli Swiss Helped to consolidate

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  16. Teleman Dan dThelema29 decembrie 2019 la 15:58

    equations using a matrix
    1654-1705 Jacob Bernoulli Swiss Helped to consolidate infinitesimal calculus, developed a technique for solving separable differential equations, added a theory of permutations and combinations to probability theory, Bernoulli Numbers sequence, transcendental curves
    1667-1748 Johann Bernoulli Swiss Further developed infinitesimal calculus, including the “calculus of variation”, functions for curve of fastest descent (brachistochrone) and catenary curve
    1667-1754 Abraham de Moivre French De Moivre's formula, development of analytic geometry, first statement of the formula for the normal distribution curve, probability theory
    1690-1764 Christian Goldbach German Goldbach Conjecture, Goldbach-Euler Theorem on perfect powers
    1707-1783 Leonhard Euler Swiss Made important contributions in almost all fields and found unexpected links between different fields, proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks
    1728-1777 Johann Lambert Swiss Rigorous proof that π is irrational, introduced hyperbolic functions into trigonometry, made conjectures on non-Euclidean space and hyperbolic triangles
    1736-1813 Joseph Louis Lagrange Italian/French Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange’s theorem of finite groups, four-square theorem, mean value theorem
    1746-1818 Gaspard Monge French Inventor of descriptive geometry, orthographic projection
    1749-1827 Pierre-Simon Laplace French Celestial mechanics

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  17. Teleman Dan dThelema29 decembrie 2019 la 15:59

    projection
    1749-1827 Pierre-Simon Laplace French Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism
    1752-1833 Adrien-Marie Legendre French Abstract algebra, mathematical analysis, least squares method for curve-fitting and linear regression, quadratic reciprocity law, prime number theorem, elliptic functions
    1768-1830 Joseph Fourier French Studied periodic functions and infinite sums in which the terms are trigonometric functions (Fourier series)
    1777-1825 Carl Friedrich Gauss German Pattern in occurrence of prime numbers, construction of heptadecagon, Fundamental Theorem of Algebra, exposition of complex numbers, least squares approximation method, Gaussian distribution, Gaussian function, Gaussian error curve, non-Euclidean geometry, Gaussian curvature
    1789-1857 Augustin-Louis Cauchy French Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory)
    1790-1868 August Ferdinand Möbius German Möbius strip (a two-

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  18. Teleman Dan dThelema29 decembrie 2019 la 16:00

    Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula
    1791-1858 George Peacock British Inventor of symbolic algebra (early attempt to place algebra on a strictly logical basis)
    1791-1871 Charles Babbage British Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer.
    1792-1856 Nikolai Lobachevsky Russian Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai
    1802-1829 Niels Henrik Abel Norwegian Proved impossibility of solving quintic equations, group theory, abelian groups, abelian categories, abelian variety
    1802-1860 János Bolyai Hungarian Explored hyperbolic geometry and curved spaces independently of Lobachevsky
    1804-1851 Carl Jacobi German Important contributions to analysis, theory of periodic and elliptic functions, determinants and matrices
    1805-1865 William Hamilton Irish Theory of quaternions (first example of a non-commutative algebra)
    1811-1832 Évariste Galois French Proved that there is no general algebraic method for solving polynomial equations of degree greater than four, laid groundwork for abstract algebra, Galois theory, group theory, ring theory, etc
    1815-1864 George Boole British Devised Boolean

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  19. Teleman Dan dThelema29 decembrie 2019 la 16:01

    Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science
    1815-1897 Karl Weierstrass German Discovered a continuous function with no derivative, advancements in calculus of variations, reformulated calculus in a more rigorous fashion, pioneer in development of mathematical analysis
    1821-1895 Arthur Cayley British Pioneer of modern group theory, matrix algebra, theory of higher singularities, theory of invariants, higher dimensional geometry, extended Hamilton's quaternions to create octonions
    1826-1866 Bernhard Riemann German Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis
    1831-1916 Richard Dedekind German Defined some important concepts of set theory such as similar sets and infinite sets, proposed Dedekind cut (now a standard definition of the real numbers)
    1834-1923 John Venn British Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics)
    1842-1899 Marius Sophus Lie Norwegian Applied algebra to

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  20. Teleman Dan dThelema29 decembrie 2019 la 16:03

    Applied algebra to geometric theory of differential equations, continuous symmetry, Lie groups of transformations
    1845-1918 Georg Cantor German Creator of set theory, rigorous treatment of the notion of infinity and transfinite numbers, Cantor's theorem (which implies the existence of an “infinity of infinities”)
    1848-1925 Gottlob Frege German One of the founders of modern logic, first rigorous treatment of the ideas of functions and variables in logic, major contributor to study of the foundations of mathematics
    1849-1925 Felix Klein German Klein bottle (a one-sided closed surface in four-dimensional space), Erlangen Program to classify geometries by their underlying symmetry groups, work on group theory and function theory
    1854-1912 Henri Poincaré French Partial solution to “three body problem”, foundations of modern chaos theory, extended theory of mathematical topology, Poincaré conjecture
    1858-1932 Giuseppe Peano Italian Peano axioms for natural numbers

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  21. Teleman Dan dThelema29 decembrie 2019 la 16:03

    Peano axioms for natural numbers, developer of mathematical logic and set theory notation, contributed to modern method of mathematical induction
    1861-1947 Alfred North Whitehead British Co-wrote “Principia Mathematica” (attempt to ground mathematics on logic)
    1862-1943 David Hilbert German 23 “Hilbert problems”, finiteness theorem, “Entscheidungsproblem“ (decision problem), Hilbert space, developed modern axiomatic approach to mathematics, formalism
    1864-1909 Hermann Minkowski German Geometry of numbers (geometrical method in multi-dimensional space for solving number theory problems), Minkowski space-time
    1872-1970 Bertrand Russell British Russell’s paradox, co-wrote “Principia Mathematica” (attempt to ground mathematics on logic), theory of types
    1877-1947 G.H. Hardy British Progress toward solving Riemann hypothesis (proved infinitely many zeroes on the critical line), encouraged new tradition of pure mathematics in Britain, taxicab numbers
    1878-1929 Pierre Fatou French Pioneer in field of complex analytic dynamics, investigated iterative and recursive processes
    1881-1966 L.E.J. Brouwer Dutch Proved several

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  22. Teleman Dan dThelema29 decembrie 2019 la 16:04

    Proved several theorems marking breakthroughs in topology (including fixed point theorem and topological invariance of dimension)
    1887-1920 Srinivasa Ramanujan Indian Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions
    1893-1978 Gaston Julia French Developed complex dynamics, Julia set formula
    1903-1957 John von Neumann Hungarian/
    American Pioneer of game theory, design model for modern computer architecture, work in quantum and nuclear physics
    1906-1978 Kurt Gödel Austria Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory
    1906-1998 André Weil French Theorems allowed connections between algebraic geometry and number theory, Weil conjectures (partial proof of Riemann hypothesis for local zeta functions), founding member of influential Bourbaki group
    1912-1954 Alan Turing British Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence
    1913-1996 Paul Erdös Hungarian Set and solved many problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory
    1917-2008 Edward Lorenz American Pioneer in modern

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  23. Teleman Dan dThelema29 decembrie 2019 la 16:05

    Pioneer in modern chaos theory, Lorenz attractor, fractals, Lorenz oscillator, coined term “butterfly effect”
    1919-1985 Julia Robinson American Work on decision problems and Hilbert's tenth problem, Robinson hypothesis
    1924-2010 Benoît Mandelbrot French Mandelbrot set fractal, computer plottings of Mandelbrot and Julia sets
    1928-2014 Alexander Grothendieck French Mathematical structuralist, revolutionary advances in algebraic geometry, theory of schemes, contributions to algebraic topology, number theory, category theory, etc
    1928-2015 John Nash American Work in game theory, differential geometry and partial differential equations, provided insight into complex systems in daily life such as economics, computing and military
    1934-2007 Paul Cohen American Proved that continuum hypothesis could be both true and not true (i.e. independent from Zermelo-Fraenkel set theory)
    1937- John Horton Conway British Important contributions to game theory, group theory, number theory, geometry and (especially) recreational mathematics, notably with the invention of the cellular automaton called the "Game of Life"
    1947- Yuri Matiyasevich Russian Final proof that Hilbert’s tenth problem is impossible (there is no general method for determining whether Diophantine equations have a solution)
    1953- Andrew Wiles British Finally proved Fermat’s Last Theorem for all

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  24. Teleman Dan dThelema29 decembrie 2019 la 16:06

    Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves)
    1966- Grigori Perelman Russian Finally proved Poincaré Conjecture (by proving Thurston's geometrization conjecture), contributions to Riemannian geometry and geometric topology

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  25. Teleman Dan dThelema29 decembrie 2019 la 16:09

    https://mathoverflow.net/questions/53122/mathematical-urban-legends

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  26. Dan Teleman13 ianuarie 2023 la 03:45

    GPT....G.P.T.

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