Erdős Conjecture On Arithmetic Progressions

	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 three-term arithmetic progressions.. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in wha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Of Coloring And The Colorful Life Of Its Creators Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove properties of objects, a ''proof'' consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]