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Odd Greedy Expansion
In number theory, the odd greedy expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. , it remains unsolved. Description An Egyptian fraction represents a given rational number as a sum of distinct unit fractions. If a rational number x/y is a sum of unit fractions with odd denominators, :\frac = \sum\frac, then y must be odd. Conversely, every fraction x/y with y odd can be represented as a sum of distinct odd unit fractions. One method of finding such a representation replaces x/y by Ax/Ay where A=35\cdot 3^i for a sufficiently large i, and then expands Ax as a sum of distinct divisors of Ay.; . However, a simpler greedy algorithm has successfully found Egyptian fractions in which all denominators are odd for all instances x/y (with odd y) on which it has been tested: let u be the least odd number that is greater than or equal to y/x, include the fraction 1/u in the expansion, and continue in the same way (avoidi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...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] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egyptian Fraction
An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number \tfrac; for instance the Egyptian fraction above sums to \tfrac. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including \tfrac and \tfrac as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics. Appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. Arithmetic Elementary arithmetic Multiplying any two unit fractions results in a product that is another unit fraction: \frac1x \times \frac1y = \frac1. However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction: \frac1x + \frac1y = \frac \frac1x - \frac1y = \frac \frac1x \div \frac1y = \frac. Modular arithmetic In modular arithmetic, unit fractions can often be converted into equivalent integers using a calculation based on greatest common divisors. In turn, this conversion can be used to simplify division operations in modular arithmetic, by transforming them into equivalent multiplication operations. Specifically, consider the problem of dividing by a value x modulo y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm For Egyptian Fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as . As the name indicates, these representations have been used as long ago as ancient Egypt, but the first published systematic method for constructing such expansions was described in 1202 in the ''Liber Abaci'' of Leonardo of Pisa (Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible unit fraction that can be used in any representation of the remaining fraction. Fibonacci actually lists several different methods for constructing Egyptian fraction representations. He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. As Salzer ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Selfridge
John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin. Career Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University from 1971 to 1991 (retirement), chairing the Department of Mathematical Sciences 1972–1976 and 1986–1990. He was executive editor of Mathematical Reviews from 1978 to 1986, overseeing the computerization of its operations. He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour. Research In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when ''k'' = 78,557, all numbers of the form ''k''2''n'' + 1 have a factor in the covering set . Five ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Graham
Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences. After graduate study at the University of California, Berkeley, Graham worked for many years at Bell Labs and later at the University of California, San Diego. He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness, and many topics in mathematics are named after him. He published six books and about 400 papers, and had nearly 200 co-authors, including many collaborative works with his wife Fan Chung and with Paul Erdős. Graham has been featured in ''Ripley's Believe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are :2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 . Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms. Formal definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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