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Erdős Conjecture On Arithmetic Progressions
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). It states that if the sum of the reciprocals of the members of a set ''A'' of positive integers diverges, then ''A'' contains arbitrarily long arithmetic progressions. Formally, the conjecture states that if ''A'' is a large set in the sense that :: \sum_ \frac \ =\ \infty, then ''A'' contains arithmetic progressions of any given length, meaning that for every positive integer ''k'' there are an integer ''a'' and a non-zero integer ''c'' such that \\subset A. History In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many 3 term arithmetic progressions.. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Conjectures By Paul Erdős
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them. Unsolved * The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3. * The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. * The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers. * The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus. * The Erdős–Straus conjecture on the Diophantine equation 4/''n'' = 1/''x'' + 1/''y'' + 1/''z''. * The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals. * The Erdős–Szekeres conjecture on the number of points needed to ensure that a point se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Sums Of Reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first ''n'' of them are summed, then one more is included to give the sum of the first ''n''+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problems Involving Arithmetic Progressions
Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view. Largest progression-free subsets Find the cardinality (denoted by ''A''''k''(''m'')) of the largest subset of which contains no progression of ''k'' distinct terms. The elements of the forbidden progressions are not required to be consecutive. For example, ''A''4(10) = 8, because has no arithmetic progressions of length 4, while all 9-element subsets of have one. Paul Erdős set a $1000 prize for a question related to this number, collected by Endre Szemerédi for what has become known as Szemerédi's theorem. Arithmetic progressions from prime numbers Szemerédi's theorem states that a set of natural numbers of non-zero upper asymptotic density contains finite arithmetic progressions, of any arbitrary length ''k''. Erdős made Erdős conjecture on arithmetic progressions, a more general conjecture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Tao Theorem
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', there exist arithmetic progressions of primes with ''k'' terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.. Statement Let \pi(N) denote the number of primes less than or equal to N. If A is a subset of the prime numbers such that : \limsup_ \frac>0, then for all positive integers k, the set A contains infinitely many arithmetic progressions of length k. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula : (\mathfrak_k + o(1))\frac for the number of ''k'' tuples of prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Coloring And The Colorful Life Of Its Creators
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
