• The Hundred Greatest Theorems


    The Hundred Greatest Theorems

    The millenium seemed to spur a lot of people to compile "Top 100" or "Best 100" lists of many things, including movies (by the American Film Institute) and books (by the Modern Library). Mathematicians were not immune, and at a mathematics conference in July, 1999, Paul and Jack Abad presented their list of "The Hundred Greatest Theorems." Their ranking is based on the following criteria: "the place the theorem holds in the literature, the quality of the proof, and the unexpectedness of the result."

    The list is of course as arbitrary as the movie and book list, but the theorems here are all certainly worthy results. I hope to over time include links to the proofs of them all; for now, you'll have to content yourself with the list itself and the biographies of the principals.

     

     

    1

    The Irrationality of the Square Root of 2

    Pythagoras and his school

    500 B.C.

    2

    Fundamental Theorem of Algebra

    Karl Frederich Gauss

    1799

    3

    The Denumerability of the Rational Numbers

    Georg Cantor

    1867

    4

    Pythagorean Theorem

    Pythagoras and his school

    500 B.C.

    5

    Prime Number Theorem

    Jacques Hadamard and Charles-Jean de la Vallee Poussin(separately)

    1896

    6

    Godel’s Incompleteness Theorem

    Kurt Godel

    1931

    7

    Law of Quadratic Reciprocity

    Karl Frederich Gauss

    1801

    8

    The Impossibility of Trisecting the Angle and Doubling the Cube

    Pierre Wantzel

    1837

    9

    The Area of a Circle

    Archimedes

    225 B.C.

    10

    Euler’s Generalization of Fermat’s Little Theorem

    (Fermat’s Little Theorem)

    Leonhard Euler

    (Pierre de Fermat)

    1760

    (1640)

    11

    The Infinitude of Primes

    Euclid

    300 B.C.

    12

    The Independence of the Parallel Postulate

    Karl Frederich Gauss, Janos Bolyai, Nikolai Lobachevsky, G.F. Bernhard Riemann collectively

    1870-1880

    13

    Polyhedron Formula

    Leonhard Euler

    1751

    14

    Euler’s Summation of 1 + (1/2)^2 + (1/3)^2 + ….

    Leonhard Euler

    1734

    15

    Fundamental Theorem of Integral Calculus

    Gottfried Wilhelm von Leibniz

    1686

    16

    Insolvability of General Higher Degree Equations

    Niels Henrik Abel

    1824

    17

    DeMoivre’s Theorem

    Abraham DeMoivre

    1730

    18

    Liouville’s Theorem and the Construction of Trancendental Numbers

    Joseph Liouville

    1844

    19

    Four Squares Theorem

    Joseph-Louis Lagrange

    1770

    20

    All Primes Equal the Sum of Two Squares

    ?

    ?

    21

    Green’s Theorem

    George Green

    1828

    22

    The Non-Denumerability of the Continuum

    Georg Cantor

    1874

    23

    Formula for Pythagorean Triples

    Euclid

    300 B.C.

    24

    The Undecidability of the Coninuum Hypothesis

    Paul Cohen

    1963

    25

    Schroeder-Bernstein Theorem

    ?

    ?

    26

    Leibnitz’s Series for Pi

    Gottfried Wilhelm von Leibniz

    1674

    27

    Sum of the Angles of a Triangle

    Euclid

    300 B.C.

    28

    Pascal’s Hexagon Theorem

    Blaise Pascal

    1640

    29

    Feuerbach’s Theorem

    Karl Wilhelm Feuerbach

    1822

    30

    The Ballot Problem

    J.L.F. Bertrand

    1887

    31

    Ramsey’s Theorem

    F.P. Ramsey

    1930

    32

    The Four Color Problem

    Kenneth Appel and Wolfgang Haken

    1976

    33

    Fermat’s Last Theorem

    Andrew Wiles

    1993

    34

    Divergence of the Harmonic Series

    Nicole Oresme

    1350

    35

    Taylor’s Theorem

    Brook Taylor

    1715

    36

    Brouwer Fixed Point Theorem

    L.E.J. Brouwer

    1910

    37

    The Solution of a Cubic

    Scipione Del Ferro

    1500

    38

    Arithmetic Mean/Geometric Mean

    (Proof by Backward Induction)

    (Polya Proof)

    Augustin-Louis Cauchy

    George Polya

    ?

    ?

    39

    Solutions to Pell’s Equation

    Leonhard Euler

    1759

    40

    Minkowski’s Fundamental Theorem

    Hermann Minkowski

    1896

    41

    Puiseux’s Theorem

    Victor Puiseux (based on a discovery of Isaac Newton of 1671)

    1850

    42

    Sum of the Reciprocals of the Triangular Numbers

    Gottfried Wilhelm von Leibniz

    1672

    43

    The Isoperimetric Theorem

    Jacob Steiner

    1838

    44

    The Binomial Theorem

    Isaac Newton

    1665

    45

    The Partition Theorem

    Leonhard Euler

    1740

    46

    The Solution of the General Quartic Equation

    Lodovico Ferrari

    1545

    47

    The Central Limit Theorem

    ?

    ?

    48

    Dirichlet’s Theorem

    Peter Lejune Dirichlet

    1837

    49

    The Cayley-Hamilton Thoerem

    Arthur Cayley

    1858

    50

    The Number of Platonic Solids

    Theaetetus

    400 B.C.

    51

    Wilson’s Theorem

    Joseph-Louis Lagrange

    1773

    52

    The Number of Subsets of a Set

    ?

    ?

    53

    Pi is Trancendental

    Ferdinand Lindemann

    1882

    54

    Konigsberg Bridges Problem

    Leonhard Euler

    1736

    55

    Product of Segments of Chords

    Euclid

    300 B.C.

    56

    The Hermite-Lindemann Transcendence Theorem

    Ferdinand Lindemann

    1882

    57

    Heron’s Formula

    Heron of Alexandria

    75

    58

    Formula for the Number of Combinations

    ?

    ?

    59

    The Laws of Large Numbers

    <many>

    <many>

    60

    Bezout’s Theorem

    Etienne Bezout

    ?

    61

    Theorem of Ceva

    Giovanni Ceva

    1678

    62

    Fair Games Theorem

    ?

    ?

    63

    Cantor’s Theorem

    Georg Cantor

    1891

    64

    L’Hopital’s Rule

    John Bernoulli

    1696?

    65

    Isosceles Triangle Theorem

    Euclid

    300 B.C.

    66

    Sum of a Geometric Series

    Archimedes

    260 B.C.?

    67

    e is Transcendental

    Charles Hermite

    1873

    68

    Sum of an arithmetic series

    Babylonians

    1700 B.C.

    69

    Greatest Common Divisor Algorithm

    Euclid

    300 B.C.

    70

    The Perfect Number Theorem

    Euclid

    300 B.C.

    71

    Order of a Subgroup

    Joseph-Louis Lagrange

    1802

    72

    Sylow’s Theorem

    Ludwig Sylow

    1870

    73

    Ascending or Descending Sequences

    Paul Erdos and G. Szekeres

    1935

    74

    The Principle of Mathematical Induction

    Levi ben Gerson

    1321

    75

    The Mean Value Theorem

    Augustine-Louis Cauchy

    1823

    76

    Fourier Series

    Joseph Fourier

    1811

    77

    Sum of kth powers

    Jakob Bernouilli

    1713

    78

    The Cauchy-Schwarz Inequality

    Augustine-Louis Cauchy

    1814?

    79

    The Intermediate Value Theorem

    Augustine-Louis Cauchy

    1821

    80

    The Fundamental Theorem of Arithmetic

    Euclid

    300 B.C.

    81

    Divergence of the Prime Reciprocal Series

    Leonhard Euler

    1734?

    82

    Dissection of Cubes (J.E. Littlewood’s ‘elegant’ proof)

    R.L. Brooks

    1940

    83

    The Friendship Theorem

    Paul Erdos, Alfred Renyi, Vera Sos

    1966

    84

    Morley’s Theorem

    Frank Morley

    1899

    85

    Divisibility by 3 Rule

    ?

    ?

    86

    Lebesgue Measure and Integration

    Henri Lebesgue

    1902

    87

    Desargues’s Theorem

    Gerard Desargues

    1650

    88

    Derangements Formula

    ?

    ?

    89

    The Factor and Remainder Theorems

    ?

    ?

    90

    Stirling’s Formula

    James Stirling

    1730

    91

    The Triangle Inequality

    ?

    ?

    92

    Pick’s Theorem

    George Pick

    1899

    93

    The Birthday Problem

    ?

    ?

    94

    The Law of Cosines

    Francois Viete

    1579

    95

    Ptolemy’s Theorem

    Ptolemy

    120?

    96

    Principle of Inclusion/Exclusion

    ?

    ?

    97

    Cramer’s Rule

    Gabriel Cramer

    1750

    98

    Bertrand’s Postulate

    J.L.F. Bertrand

    1860?

    99

    Buffon Needle Problem

    Comte de Buffon

    1733

    100

    Descartes Rule of Signs

    Rene Descartes

    1637

    转载自 http://www.math.org.cn/forum.php?mod=viewthread&tid=31920

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  • 原文地址:https://www.cnblogs.com/zhangwenbiao/p/4198403.html
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