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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] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumset
In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is, :A + B = \. The n-fold iterated sumset of A is :nA = A + \cdots + A, where there are n summands. Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form :4\,\Box = \mathbb, where \Box is the set of square numbers. A subject that has received a fair amount of study is that of sets with ''small doubling'', where the size of the set A+A is small (compared to the size of A); see for example Freiman's theorem. See also *Restricted sumset * Sidon set *Sum-free set * Schnirelmann density *Shapley–Folkman lemma *X + Y sorting X, or x, is the twenty-fourth Letter (alphabet), letter of the Latin alphab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moser–de Bruijn Sequence
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4. Equivalently, they are the numbers whose binary representations are nonzero only in even positions. The ''Moser–de Bruijn numbers'' in this sequence grow in proportion to the square numbers. They are the squares for a modified form of arithmetic without carrying. The difference of two Moser–de Bruijn numbers, multiplied by two, is never square. Every natural number can be formed in a unique way as the sum of a Moser–de Bruijn number and twice a Moser–de Bruijn number. This representation as a sum defines a one-to-one correspondence between integers and pairs of integers, listed in order of their positions on a Z-order curve. The Moser–de Bruijn sequence can be used to construct pairs of transcendental numbers that are multiplicative inverses of each other and both have simple decimal representati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb Ruler
In mathematics, a Golomb ruler is a set (mathematics), set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its ''order'', and the largest distance between two of its marks is its ''length''. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays. The Golomb ruler was named for Solomon W. Golomb and discovered independently by and . Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruence (geometry), congruent. This turned out to be false for six-point rulers, but true otherwise. There is no requirement that a Golomb ruler be able to measure ''all'' distances up to its length, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vera T
Vera may refer to: Names *Vera (surname), a surname (including a list of people with the name) *Vera (given name), a given name (including a list of people and fictional characters with the name) **Vera (), archbishop of the archdiocese of Tarragona Places Spain *Vera, Almería, a municipality in the province of Almería, Andalusia * Vera de Bidasoa, a municipality in the autonomous community of Navarra *La Vera, a comarca in the province of Cáceres, Extremadura United States * Vera, Illinois, an unincorporated community * Vera, Kansas, a ghost town * Vera, Missouri, an unincorporated community * Vera, Oklahoma, a town * Vera, Texas, an unincorporated community * Vera, Virginia, an unincorporated community * Veradale, Washington, originally known as Vera, CDP Elsewhere * Vera, Santa Fe, a city in the province of Santa Fe, Argentina * Vera Department, an administrative subdivision (departamento) of the province of Santa Fe * Vera, Mato Grosso, Brazil, a municipality * Cape Ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
András Sárközy
András Sárközy (born in Budapest) is a Hungarian mathematician, working in analytic and combinatorial number theory, although his first works were in the fields of geometry and classical analysis. He has the largest number of papers co-authored with Paul Erdős (a total of 62); he has an Erdős number of one. He proved the Furstenberg–Sárközy theorem that every sequence of natural numbers with positive upper density contains two members whose difference is a full square. He was elected a corresponding member (1998), and a full member (2004) of the Hungarian Academy of Sciences. He received the Széchenyi Prize The Széchenyi Prize (), named after István Széchenyi, is a prize given in Hungary by the state, replacing the former State Prize in 1990 in recognition of those who have made an outstanding contribution to academic life in Hungary. Recipients ... (2010). He is the father of the mathematician Gábor N. Sárközy. References * {{DEFAULTSORT:Sarkozy, Andr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2021 impact factor of 0.979. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Sum Of Powers Conjecture
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is itself a th power, then is greater than or equal to : a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case : if a_1^k + a_2^k = b^k, then . Although the conjecture holds for the case (which follows from Fermat's Last Theorem for the third powers), it was disproved for and . It is unknown whether the conjecture fails or holds for any value . Background Euler was aware of the equality involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Part
In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, for floor: , , and for ceiling: , and . The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). However, the same term, ''integer part'', is also used for truncation towards zero, which differs from the floor function for negative numbers. For an integer , . Although and produce graphs that appear exactly alike, they are not the same when the value of is an exact integer. For example, when , . However, if , then , while . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
