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 |
Pythagoras and his school |
500 B.C. |
|
2 |
1799 |
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3 |
1867 |
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4 |
Pythagoras and his school |
500 B.C. |
|
5 |
Jacques Hadamard and Charles-Jean de la Vallee Poussin (separately) |
1896 |
|
6 |
Godel’s Incompleteness Theorem |
1931 |
|
7 |
1801 |
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8 |
The Impossibility of Trisecting the Angle and Doubling the Cube |
1837 |
|
9 |
225 B.C. |
||
10 |
Euler’s Generalization of Fermat’s Little Theorem |
1760 (1640) |
|
11 |
300 B.C. |
||
12 |
Karl Frederich Gauss, Janos Bolyai, Nikolai Lobachevsky, G.F. Bernhard Riemann collectively |
1870-1880 |
|
13 |
1751 |
||
14 |
Euler’s Summation of 1 + (1/2)^2 + (1/3)^2 + …. |
1734 |
|
15 |
1686 |
||
16 |
Insolvability of General Higher Degree Equations |
1824 |
|
17 |
1730 |
||
18 |
Liouville’s Theorem and the Construction of Trancendental Numbers |
1844 |
|
19 |
1770 |
||
20 |
All Primes Equal the Sum of Two Squares |
? |
? |
21 |
1828 |
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22 |
The Non-Denumerability of the Continuum |
1874 |
|
23 |
Formula for Pythagorean Triples |
300 B.C. |
|
24 |
The Undecidability of the Coninuum Hypothesis |
1963 |
|
25 |
Schroeder-Bernstein Theorem |
? |
? |
26 |
Leibnitz’s Series for Pi |
1674 |
|
27 |
Sum of the Angles of a Triangle |
300 B.C. |
|
28 |
Pascal’s Hexagon Theorem |
1640 |
|
29 |
Feuerbach’s Theorem |
1822 |
|
30 |
The Ballot Problem |
1887 |
|
31 |
Ramsey’s Theorem |
1930 |
|
32 |
The Four Color Problem |
Kenneth Appel and Wolfgang Haken |
1976 |
33 |
Fermat’s Last Theorem |
1993 |
|
34 |
Divergence of the Harmonic Series |
1350 |
|
35 |
Taylor’s Theorem |
1715 |
|
36 |
Brouwer Fixed Point Theorem |
1910 |
|
37 |
The Solution of a Cubic |
1500 |
|
38 |
Arithmetic Mean/Geometric Mean (Proof by Backward Induction) (Polya Proof) |
? ? |
|
39 |
Solutions to Pell’s Equation |
1759 |
|
40 |
Minkowski’s Fundamental Theorem |
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 |
1672 |
|
43 |
The Isoperimetric Theorem |
1838 |
|
44 |
The Binomial Theorem |
1665 |
|
45 |
The Partition Theorem |
1740 |
|
46 |
The Solution of the General Quartic Equation |
1545 |
|
47 |
The Central Limit Theorem |
? |
? |
48 |
Dirichlet’s Theorem |
1837 |
|
49 |
The Cayley-Hamilton Thoerem |
1858 |
|
50 |
The Number of Platonic Solids |
400 B.C. |
|
51 |
Wilson’s Theorem |
1773 |
|
52 |
The Number of Subsets of a Set |
? |
? |
53 |
Pi is Trancendental |
1882 |
|
54 |
Konigsberg Bridges Problem |
1736 |
|
55 |
Product of Segments of Chords |
300 B.C. |
|
56 |
The Hermite-Lindemann Transcendence Theorem |
1882 |
|
57 |
Heron’s Formula |
75 |
|
58 |
Formula for the Number of Combinations |
? |
? |
59 |
The Laws of Large Numbers |
<many> |
<many> |
60 |
Bezout’s Theorem |
? |
|
61 |
Theorem of Ceva |
1678 |
|
62 |
Fair Games Theorem |
? |
? |
63 |
Cantor’s Theorem |
1891 |
|
64 |
L’Hopital’s Rule |
1696? |
|
65 |
Isosceles Triangle Theorem |
300 B.C. |
|
66 |
Sum of a Geometric Series |
260 B.C.? |
|
67 |
e is Transcendental |
1873 |
|
68 |
Sum of an arithmetic series |
Babylonians |
1700 B.C. |
69 |
Greatest Common Divisor Algorithm |
300 B.C. |
|
70 |
The Perfect Number Theorem |
300 B.C. |
|
71 |
Order of a Subgroup |
1802 |
|
72 |
Sylow’s Theorem |
1870 |
|
73 |
Ascending or Descending Sequences |
Paul Erdos and G. Szekeres |
1935 |
74 |
The Principle of Mathematical Induction |
1321 |
|
75 |
The Mean Value Theorem |
1823 |
|
76 |
Fourier Series |
1811 |
|
77 |
Sum of kth powers |
1713 |
|
78 |
The Cauchy-Schwarz Inequality |
1814? |
|
79 |
The Intermediate Value Theorem |
1821 |
|
80 |
The Fundamental Theorem of Arithmetic |
300 B.C. |
|
81 |
Divergence of the Prime Reciprocal Series |
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 |
1899 |
|
85 |
Divisibility by 3 Rule |
? |
? |
86 |
Lebesgue Measure and Integration |
1902 |
|
87 |
Desargues’s Theorem |
1650 |
|
88 |
Derangements Formula |
? |
? |
89 |
The Factor and Remainder Theorems |
? |
? |
90 |
Stirling’s Formula |
1730 |
|
91 |
The Triangle Inequality |
? |
? |
92 |
George Pick |
1899 |
|
93 |
The Birthday Problem |
? |
? |
94 |
The Law of Cosines |
1579 |
|
95 |
Ptolemy’s Theorem |
120? |
|
96 |
Principle of Inclusion/Exclusion |
? |
? |
97 |
Cramer’s Rule |
1750 |
|
98 |
Bertrand’s Postulate |
1860? |
|
99 |
Buffon Needle Problem |
1733 |
|
100 |
Descartes Rule of Signs |
1637 |