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

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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