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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.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
The first complete proofs were given by Marcel-Paul Schützenberger in 1977 and Thomas in 1974. Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In ...
Since the proofs generated by automated theorem provers are typically very large, the problem of proof compression is crucial, and various techniques aiming at making the prover's output smaller, and consequently more easily understandable and checkable, have been developed. Proof assistants require a human user to give hints to the system ...
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 ...
An archetypal double counting proof is for the well known formula for the number () of k-combinations (i.e., subsets of size k) of an n-element set: = (+) ().Here a direct bijective proof is not possible: because the right-hand side of the identity is a fraction, there is no set obviously counted by it (it even takes some thought to see that the denominator always evenly divides the numerator).
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This term reflects the common practice of bracketing each linear equation. [15] The complexity index is the number of integrand sums minus the number of brackets (linear equations). Each series expansion of the integrand contributes one sum. [15] The summation indices (variables) are the indices that index terms in a series expansion.