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List Of Long Proofs
This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. , the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full. Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. *1799 The Abel–Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. *1890 Killing's classification of simple complex Lie algebras, including his discovery of the exceptional Lie algebras, took 180 pages in 4 papers. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Adian
Sergei Ivanovich Adian, also Adyan ( hy, Սերգեյ Իվանովիչ Ադյան; russian: Серге́й Ива́нович Адя́н; 1 January 1931 – 5 May 2020), 4381, and hence for all multiples of those odd integers as well. The solution of the Burnside problem was certainly one of the most outstanding and deep mathematical results of the past century. At the same time, this result is one of the hardest theorems: just the inductive step of a complicated induction used in the proof took up a whole issue of volume 32 of Izvestiya, even lengthened by 30 pages. In many respects the work was literally carried to its conclusion by the exceptional persistence of Adian. In that regard it is worth recalling the words of Novikov, who said that he had never met a mathematician more ‘penetrating’ than Adian. In contrast to the Adian–Rabin theorem, the paper of Adian and Novikov in no way ‘closed’ the Burnside problem. Moreover, over a long period of more than ten years Adi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trichotomy Theorem
In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (al ... and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups. References * * * Theorems about finite groups {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Eisenstein series for the modular group Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: :G_(\tau) = \sum_ \frac. This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Relation to modular invariants The modular invariants and of an elliptic curve are given by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorenstein–Harada Theorem
In mathematical finite group theory, the Gorenstein–Harada theorem, proved by in a 464-page paper, ''Medium'', Cami Rosso, Feb 23, 2017 classifies the simple finite groups of sectional 2-rank at most 4. It is part of the . Finite simple groups of section 2 that rank at least 5, have s with a self-centralizing |
4-color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubters remain. The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map). To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety over a finite field with elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers of points over the extension field with elements. Weil conjectured that such ''zeta functions'' for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Order Theorem
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing XML schemas * Oodnadatta Airport (IATA: ODD), South Australia * Oppositional defiant disorder, a mental disorder characterized by anger-guided, hostile behavior * Operational due diligence * Operational Design Domain (ODD) in case of autonomous cars * Optical disc drive * ''ODD'', a 2007 play by Hal Corley about a teenager with oppositional defiant disorder Mathematics * Even and odd numbers, an integer is odd if dividing by two does not yield an integer * Even and odd functions, a function is odd if ''f''(−''x'') = −''f''(''x'') for all ''x'' * Even and odd permutations, a permutation of a finite set is odd if it is composed of an odd number of transpositions Ships * HNoMS ''Odd'', a Storm-class patrol boat of the Royal No ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-group Theorem
In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial ''p''-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups. Simple N-groups The simple N-groups were classified by in a series of 6 papers totaling about 400 pages. The simple N-groups consist of the special linear groups PSL2(''q''), PSL3(3), the Suzuki groups Sz(22''n''+1), the unitary group U3(3), the alternating group ''A''7, the Mathieu group M11, and the Tits group. (The Tits group was overlooked in Thomson's original announcement in 1968, but Hearn pointed out that it was also a simple N-group.) More generally Thompson showed that any non-solvable N-group is a subgroup of Aut(''G'') containing ''G'' for some simple N-group ''G''. generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable. The only extra simple groups t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
