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A two-column proof published in 1913. A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. [29] The proof is written as a series of lines in two columns.
In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). [1] The table below lists Gardner's columns. [2] Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "[cover]" in the table with a hyperlink to the cover. [3]
Appel and Haken's proof of this took 139 pages, and also depended on long computer calculations. 1974 The Gorenstein–Harada theorem classifying finite groups of sectional 2-rank at most 4 was 464 pages long. 1976 Eisenstein series. Langlands's proof of the functional equation for Eisenstein series was 337 pages long. 1983 Trichotomy theorem ...
The Pythagorean theorem has at least 370 known proofs. [1]In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. [a] [2] [3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Because each column has an even number of points (namely q−1 points), the number of such lattice points in the region BCYX is the same modulo 2 as the number of such points in the region CZY: The number of points with even x-coordinate inside BCYX (marked by O's) is equal modulo 2 to the number of such points in CZY (marked by X's)
The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...
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