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Erdős–Woods Number
In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property: there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, is an Erdős–Woods number if there exists a positive integer such that for each integer between and , at least one of the greatest common divisors or is greater than . Examples The first Erdős–Woods numbers are : 16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, 106, 112, 116 … . History Investigation of such numbers stemmed from the following prior conjecture by Paul Erdős: :There exists a positive integer such that every integer is uniquely determined by the list of prime divisors of . Alan R. Woods investigated this question for his 1981 thesis. Woods conjectured that whenever , the interval always includes a number coprime to both en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
86 (number)
86 (eighty-six) is the natural number following 85 (number), 85 and preceding 87 (number), 87. In mathematics 86 is: * nontotient and a noncototient. * the 25th distinct semiprime and the 13th of the form (2×q). * an Erdős–Woods number, since it is possible to find sequences of 86 consecutive integers such that each inner member shares a factor with either the first or the last member. * a happy number and a self number in base 10. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these). It is conjectured that 86 is the largest n for which the decimal expansion of 2n contains no 0. 86 = (8 × 6 = 48) + (4 × 8 = 32) + (3 × 2 = 6). That is, 86 is equal to the sum of the numbers formed in calculating its Persistence of a number, multiplicative persistence. In science * 86 is the atomic number of radon. * There are 86 metals on the modern periodic table. In other fields *In American English, and particularly in the food ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Set
In computability theory, a set of natural numbers is called computable, recursive, or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not. A set which is not computable is called noncomputable or undecidable. A more general class of sets than the computable ones consists of the computably enumerable (c.e.) sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number ''is'' in the set; the algorithm may give no answer (but not the wrong answer) for numbers not in the set. Formal definition A subset S of the natural numbers is called computable if there exists a total computable function f such that f(x)=1 if x\in S and f(x)=0 if x\notin S. In other words, the set S is computable if and only if the indicator function \mathbb_ is computable. E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...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] |
116 (number)
116 (one hundred ndsixteen) is the natural number following 115 and preceding 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function. 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also *116 (other) 116 (''one hundred and sixteen'') may refer to: *116 (number) *AD 116 *116 BC *116 (Devon and Cornwall) Engineer Regiment, Royal Engineers, a military unit *116 (MBTA bus) *116 (New Jersey bus) *116 (hip hop group), a Christian hip hop collective * ... References {{DEFAULT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
112 (number)
112 (one hundred ndtwelve) is the natural number following 111 and preceding 113. Mathematics 112 is an abundant number, a heptagonal number, and a Harshad number. 112 is the number of connected graphs on 6 unlabeled nodes. If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices. This is the smallest possible side length of an equilateral triangle that contains a point at integer distances from the vertices.Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, ...'' London: Penguin Group. (1987), page 119 See also * 112 (other) References {{Integers, 1 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
106 (number)
106 (one hundred ndsix) is the natural number following 105 and preceding 107. In mathematics 106 is a centered pentagonal number, a centered heptagonal number A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for ''n'' is given by ..., and a regular 19-gonal number. There are 106 mathematical trees with ten vertices. See also * 106 (other) References {{DEFAULTSORT:106 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
96 (number)
96 (ninety-six) is the natural number following 95 and preceding 97. It is a number that appears the same when turned upside down. In mathematics 96 is: * an octagonal number. * a refactorable number. * an untouchable number. * a semiperfect number since it is a multiple of 6. * an abundant number since the sum of its proper divisors is greater than 96. * the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being 24. * the sum of Euler's totient function φ(''x'') over the first seventeen integers. * strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195). * palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195). * an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member. * divisible by the number of prime numbers (24) below 96. Every integer greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
94 (number)
94 (ninety-four) is the natural number following 93 and preceding 95. In mathematics 94 is: *the twenty-ninth distinct semiprime and the fourteenth of the form (2.q). *the ninth composite number in the 43-aliquot tree. The aliquot sum of 94 is 50 within the aliquot sequence (94,50,43,1,0). *the second number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95. *a 17- gonal number and a nontotient. *an Erdős–Woods number, since it is possible to find sequences of 94 consecutive integers such that each inner member shares a factor with either the first or the last member. *a Smith number in decimal. In computing The ASCII character set (and, more generally, ISO 646) contains exactly 94 graphic non- whitespace characters, which form a contiguous range of code points. These codes ( 0x21–0x7E, as corresponding high bit set bytes 0xA1–0xFE) also used in various multi-byte encoding schemes for languages of East Asia, such as ISO 2022, EUC and GB 2312. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
