## Δευτέρα 15 Οκτωβρίου 2012

### The Hundred Greatest Theorems

 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 + .. (the Basel Problem). 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 Primes that Equal to the Sum of Two Squares (Genus theorem) ? ? 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 Continuum 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 Newtonof 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 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