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The seven selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among ...
This is a list of unusually long mathematical proofs.Such proofs often use computational proof methods and may be considered non-surveyable.. As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages.
Italian school of algebraic geometry. Most gaps in proofs are caused either by a subtle technical oversight, or before the 20th century by a lack of precise definitions. A major exception to this is the Italian school of algebraic geometry in the first half of the 20th century, where lower standards of rigor gradually became acceptable.
Pages in category "Unsolved problems in geometry" The following 48 pages are in this category, out of 48 total. This list may not reflect recent changes. A.
Turing's proof is made difficult by number of definitions required and its subtle nature. See Turing machine and Turing's proof for details. Turing's first proof (of three) follows the schema of Richard's paradox: Turing's computing machine is an algorithm represented by a string of seven letters in a "computing machine".
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
In which case, if P 1 (S) is the set of one-element subsets of S and f is a proposed bijection from P 1 (S) to P(S), one is able to use proof by contradiction to prove that |P 1 (S)| < |P(S)|. The proof follows by the fact that if f were indeed a map onto P(S), then we could find r in S, such that f({r}) coincides with the modified diagonal set ...
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
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