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Erdős–Turán Conjecture On Additive Bases
The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. It concerns additive bases, subsets of natural numbers with the property that every natural number can be represented as the sum of a bounded number of elements from the basis. Roughly, it states that the number of representations of this type cannot also be bounded. Background and formulation The question concerns subsets of the natural numbers, typically denoted by \mathbb , called additive bases. A subset B is called an (asymptotic) additive basis of finite order if there is some positive integer h such that every sufficiently large natural number n can be written as the sum of at most h elements of B. For example, the natural numbers are themselves an additive basis of order 1, since every natural number is trivially a sum of at most one natural number. Lagrange's four-square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets and of elements from an abelian group , :A + B = \, and the -fold sumset of , :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, ... [...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 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] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets and of elements from an abelian group , :A + B = \, and the -fold sumset of , :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van H
A van is a type of road vehicle used for transporting goods or people. There is some variation in the scope of the word across the different English-speaking countries. The smallest vans, microvans, are used for transporting either goods or people in tiny quantities. Mini MPVs, compact MPVs, and MPVs are all small vans usually used for transporting people in small quantities. Larger vans with passenger seats are used for institutional purposes, such as transporting students. Larger vans with only front seats are often used for business purposes, to carry goods and equipment. Specially equipped vans are used by television stations as mobile studios. Postal services and courier companies use large step vans to deliver packages. Word origin and usage Van meaning a type of vehicle arose as a contraction of the word caravan. The earliest records of a van as a vehicle in English are in the mid-19th century, meaning a covered wagon for transporting goods; the earliest reported r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Tetali Theorem
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive basis, additive bases of every order. More specifically, it states that for every fixed integer h \geq 2, there exists a subset of the natural numbers \mathcal \subseteq \mathbb satisfying r_(n) \asymp \log n, where r_(n) denotes the number of ways that a natural number ''n'' can be expressed as the sum of ''h'' elements of ''B''. The theorem is named after Paul Erdős and Prasad V. Tetali, who published it in 1990. Motivation The original motivation for this result is attributed to a problem posed by S. Sidon in 1932 on ''economical bases''. An additive basis \mathcal\subseteq\mathbb is called ''economical'' (or sometimes ''thin'') when it is an additive basis of order ''h'' and :r_(n) \ll_ n^\varepsilon for every \varepsilon > 0. In other words, these are additive bases that use as few numbers as possible to represent a given ''n'', and yet repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prasad V
200px, ''Naivedya'' offered to Sri Maya Chandrodaya Mandir in Mayapur, India">Mayapur.html" ;"title="Sri Maya Chandrodaya Mandir, Mayapur">Sri Maya Chandrodaya Mandir in Mayapur">Sri Maya Chandrodaya Mandir, Mayapur">Sri Maya Chandrodaya Mandir in Mayapur, India Prasāda (, Sanskrit: प्रसाद), prasad or prasadam is a religious offering in Hinduism. Most often ''Prasada'' is vegetarian food especially cooked for devotees after praise and thanksgiving to a god. ''Mahaprasada'' (also called ''bhandarā''),Pashaura Singh, Louis E. Fenech, 2014The Oxford Handbook of Sikh Studies/ref> is the consecrated food offered to the deity in a Hindu temple which is then distributed and partaken by all the devotees regardless of any orientation.Chitrita Banerji, 2010Eating India: Exploring the Food and Culture of the Land of SpicesSubhakanta Behera, 2002Construction of an identity discourse: Oriya literature and the Jagannath lovers (1866–1936) pp. 140–177.Susan Pattinson, 2011Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Wirsing
Eduard Wirsing (28 June 1931 – 22 March 2022) was a German mathematician, specializing in number theory. Biography Wirsing was born on 28 June 1931 in Berlin. Wirsing studied at the University of Göttingen and the Free University of Berlin, where he received his doctorate in 1957 under the supervision of Hans-Heinrich Ostmann with thesis ''Über wesentliche Komponenten in der additiven Zahlentheorie'' (On Essential Components in Additive Number Theory). In 1967/68 he was a professor at Cornell University and from 1969 a full professor at the University of Marburg, where he was since 1965. In 1970/71 he was at the Institute for Advanced Study. Since 1974 he was a professor at the University of Ulm, where he led the 1976 Mathematical Colloquium. He retired as professor emeritus in 1999, but continued to be mathematically active. Wirsing organized conferences on analytical number theory at the Oberwolfach Research Institute for Mathematics. In his spare time he played Go (game), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sidon Sequence
In number theory, a Sidon sequence is a sequence A=\ of natural numbers in which all pairwise sums a_i+a_j are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian mathematician Simon Sidon, who introduced the concept in his investigations of Fourier series. The main problem in the study of Sidon sequences, posed by Sidon, is to find the maximum number of elements that a Sidon sequence can contain, up to some bound x. Despite a large body of research, the question has remained unsolved. Early results Paul Erdős and Pál Turán proved that, for every x>0, the number of elements smaller than x in a Sidon sequence is at most \sqrt+O(\sqrt . Several years earlier, James Singer had constructed Sidon sequences with \sqrt(1-o(1)) terms less than ''x''. The upper bound was improved to \sqrt+\sqrt 1 in 1969 and to \sqrt+0.998\sqrt /math> in 2023. In 1994 Erdős offered 500 dollars for a proof or disproof of the bound \sqrt+o(x^\varepsilon). Dense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem ( Magnes Press). History Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.754. External links * Mathematics journals Academic journals established in 1963 Academic journals of Israel English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence (x_n) is denoted by \liminf_x_n\quad\text\quad \varliminf_x_n, and the limit superior of a sequence (x_n) is deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
