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Catalan's Conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the ''only'' case of two consecutive perfect powers. That is to say, that History The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case ''b'' = 2. In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on ''a'',''b'' and used existing results bounding ''x'',''y'' in terms of ''a' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. 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. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper And Lower Bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . and other numbers ''x'' such that would be an upper bound for ''S''. The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan–Nagell Equation
In number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6. Equation and solution The equation is :2^n-7=x^2 \, and solutions in natural numbers ''n'' and ''x'' exist just when ''n'' = 3, 4, 5, 7 and 15 . This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of ''n'' correspond to the values of ''x'' as:- :''x'' = 1, 3, 5, 11 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mordell Curve
In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in .... These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He showed that every Mordell curve contains only finitely many integer points (''x'', ''y''). In other words, the differences of perfect squares and perfect cubes tend to infinity. The question of how fast was dealt with in principle by Baker's method. Hypothetically this issue is dealt with by Marshall Hall's conjecture. Properties *If (''x'', ''y'') is an integer point on a Mordell curve, then so is (''x'', −''y''). *If (''x'', ''y'') is a rational point on a Mordell curve with ''y'' � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat–Catalan Conjecture
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation has only finitely many solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with distinct triplets of values (''a''''m'', ''b''''n'', ''c''''k'') where ''a'', ''b'', ''c'' are positive coprime integers and ''m'', ''n'', ''k'' are positive integers satisfying The inequality on ''m'', ''n'', and ''k'' is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with ''k'' = 1 (for any ''a'', ''b'', ''m'', and ''n'' and with ''c'' = ''a''''m'' + ''b''''n''), with ''m''=''n''=''k''=2 (for the infinitely many Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation X^y = Y^x
In general, exponentiation fails to be Commutative property, commutative. However, the equation x^y = y^x has an infinity of solutions, consisting of the line and a smooth curve intersecting the line at , where is Euler's number. The only integer solution that is on the curve is . History The equation x^y=y^x is mentioned in a letter of Daniel Bernoulli, Bernoulli to Christian Goldbach, Goldbach (29 June 1728). The letter contains a statement that when x\ne y, the only solutions in natural numbers are (2, 4) and (4, 2), although there are infinitely many solutions in rational numbers, such as (\tfrac, \tfrac) and (\tfrac, \tfrac). The reply by Goldbach (31 January 1729) contains a general solution of the equation, obtained by substituting y=vx. A similar solution was found by Leonhard Euler, Euler. J. van Hengel pointed out that if r, n are positive integers with r \geq 3, then r^ > (r+n)^r; therefore it is enough to consider possibilities x = 1 and x = 2 in order to find s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beal's Conjecture
The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in positive integers and pairwise coprime integers ''A, B, C'' if ''x, y, z'' > 2. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million. In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture. Related examples To illustrate, the solution 3^3 + 6^3 = 3^5 has bases with a commo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABC Conjecture
ABC are the first three letters of the Latin script. ABC or abc may also refer to: Arts, entertainment and media Broadcasting * Aliw Broadcasting Corporation, Philippine broadcast company * American Broadcasting Company, a commercial American TV broadcaster ** Disney–ABC Television Group, the former name of the parent organization of ABC * Australian Broadcasting Corporation, one of the national publicly funded broadcasters of Australia ** ABC Television (Australian TV network), the national television network of the Australian Broadcasting Corporation *** ABC TV (Australian TV channel), the flagship TV station of the Australian Broadcasting Corporation *** ABC Canberra (TV station), Canberra, and other ABC TV local stations in state capitals *** ABC Australia (Southeast Asian TV channel), an international pay TV channel * ABC Radio (other), several radio stations * Associated Broadcasting Corporation, the former name of TV5 Network, Inc., a Philippine media com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Bilu
Yuri may refer to: People Given name *Yuri (Slavic name), the Slavic masculine form of the given name George, including a list of people with the given name Yuri, Yury, etc. *Yuri (Japanese name), feminine Japanese given names, including a list of people and fictional characters * Yu-ri (Korean name), Korean unisex given name, including a list of people and fictional characters Mononym Singers *Yuri (Japanese singer), vocalist of the band Move * Yuri (Korean singer), member of Girl Friends *Yuri (Mexican singer) Footballers *Yuri (footballer, born 1982), full name Yuri de Souza Fonseca, Brazilian football forward * Yuri (footballer, born 1984), full name Yuri Adriano Santos, Brazilian footballer * Yuri (footballer, born 1986), full name Yuri Vera Cruz Erbas, Brazilian footballer * Yuri (footballer, born 1989), full name Yuri Naves Roberto, Brazilian football defensive midfielder * Yuri (footballer, born 1990), full name Yuri Savaroni Batista da Silva, Brazilian footballer * Yuri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ''v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
