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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] |
De Bruijn Sequence
In combinatorics, combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet (computer science), alphabet ''A'' is a cyclic sequence in which every possible length-''n'' String (computer science)#Formal theory, string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subsequence). Such a sequence is denoted by and has length , which is also the number of distinct strings of length ''n'' on ''A''. Each of these distinct strings, when taken as a substring of , must start at a different position, because substrings starting at the same position are not distinct. Therefore, must have ''at least'' symbols. And since has ''exactly'' symbols, de Bruijn sequences are optimally short with respect to the property of containing every string of length ''n'' at least once. The number of distinct de Bruijn sequences is :\dfrac. For a binary alphabet this is 2^, leading to the following sequence for positive n: 1, 1, 2, 16, 2048, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negabinary
A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base is equal to for some natural number (). Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a minus sign (or, in computer representation, a sign bit); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent. The common names for negative-base positional numeral systems are formed by prefixing ''nega-'' to the name of the corresponding positive-base system; for example, negadecimal (base −10) corresponds to decim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtract A Square
Subtract-a-square (also referred to as take-a-square) is a two-player mathematical subtraction game. It is played by two people with a pile of coins (or other tokens) between them. The players take turns removing coins from the pile, always removing a non-zero square number of coins. The game is usually played as a '' normal play'' game, which means that the player who removes the last coin wins. It is an impartial game, meaning that the set of moves available from any position does not depend on whose turn it is. Solomon W. Golomb credits the invention of this game to Richard A. Epstein.. Example A normal play game starting with 13 coins is a win for the first player provided they start with a subtraction of 1: player 1: 13 - 1*1 = 12 Player 2 now has three choices: subtract 1, 4 or 9. In each of these cases, player 1 can ensure that within a few moves the number 2 gets passed on to player 2: player 2: 12 - 1*1 = 11 player 2: 12 - 2*2 = 8 player 2: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtraction Game
In combinatorial game theory, a subtraction game is an abstract strategy game whose state can be represented by a natural number or vector of numbers (for instance, the numbers of game tokens in piles of tokens, or the positions of pieces on board) and in which the allowed moves reduce these numbers., "Subtraction games", pp. 83–86. Often, the moves of the game allow any number to be reduced by subtracting a value from a specified ''subtraction set'', and different subtraction games vary in their subtraction sets. These games also vary in whether the last player to move wins (the normal play convention) or loses (misère play convention). Another winning convention that has also been used is that a player who moves to a position with all numbers zero wins, but that any other position with no moves possible is a draw. Examples Examples of notable subtraction games include the following: * Nim is a game whose state consists of multiple piles of tokens, such as coins or matchsticks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitwise Operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the central processing unit, processor. Most bitwise operations are presented as two-operand instructions where the result replaces one of the input operands. On simple low-cost processors, typically, bitwise operations are substantially faster than division, several times faster than multiplication, and sometimes significantly faster than addition. While modern processors usually perform addition and multiplication just as fast as bitwise operations due to their longer instruction pipelines and other computer architecture, architectural design choices, bitwise operations do commonly use less power because of the reduced use of resources. Bitwise operators In the explanations below, any indication of a bit's p ... [...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] |
Imre Z
Imre () is a Hungarian masculine first name, which is also in Estonian use, where the corresponding name day is 10 April. It has been suggested that it relates to the name Emeric, Emmerich or Heinrich. Its English equivalents are Emery and Henry. Bearers of the name include the following (who generally held Hungarian nationality, unless otherwise noted): * Imre Antal (1935–2008), pianist * Imre Bajor (1957–2014), actor * Imre Bebek (d. 1395), baron * Imre Bródy (1891–1944), physicist * Imre Bujdosó (b. 1959), Olympic fencer * Imre Csáky (cardinal) (1672–1732), Roman Catholic cardinal * Imre Csermelyi (b. 1988), football player *Imre Cseszneky (1804–1874), agriculturist and patriot * Imre Csiszár (b. 1938), mathematician * Imre Csösz (b. 1969), Olympic judoka * Imre Czobor (1520–1581), Noble and statesman *Imre Czomba (b. 1972), Composer and musician * Imre Deme (b. 1983), football player * Imre Erdődy (1889–1973), Olympic gymnast * Imre Farkas (1879–1976 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Furstenberg–Sárközy Theorem
In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large. In the game of subtract a square, the positions where the next player loses form a square-difference-free set. Another square-difference-free set is obtained by doubling the Moser–de Bruijn sequence. The best known upper bound on the size of a square-difference-free set of numbers up to n is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size \approx n^. Closing the gap between these upper and lower bounds remains an open problem. The sublinear size bounds on square-difference-free sets can be generalized to sets where certain other polynomials are forbidden as differences between pairs of elements. Example An example of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
