In mathematics , an integersequence is a sequence i.e., an ordered list of integer s. An integersequence ..., an integersequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number , even though we do not have a formula for the n th perfect number. Examples Integer sequences .... Complete sequences An integersequence is called a complete sequence if every positive integer can ... between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, the Fibonacci numbers Fibonacci sequence is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one an implicit description. The sequence 0, 3, 8, 15, is formed according ... number s Baum Sweet sequence Bell number s Binomial coefficient s Carmichael number s Catalan number ... number s Fibonacci word Figurate numbers Golomb sequence Happy number s Highly totient number s Highly composite number s Home prime s Hyperperfect number s Juggler sequence Kolakoski sequence Lucky number s Lucas number s Padovan sequence Padovan number s Partition number s Perfect number s Pseudoperfect number s Prime number s Pseudoprime numbers Regular paperfolding sequence Rudin Shapiro sequence Semiperfect number s Semiprime numbers Superperfect number s Thue Morse sequence Ulam numbers Weird number s div Computable and definable sequences An integersequence is a Recursion theory computable sequence , if there exists an algorithm which given n , calculates a sub n sub , for all n > 0. An integersequence is a definable set definable sequence , if there exists some statement P x which is true for that integersequence x and false for all other integer sequences. The set of computable integer sequences and definable integer sequences are both countable , with the computable ... but not computable . The set of all integer sequences is uncountable with cardinality equal to beth ... more details
In mathematics , an integersequence prime is a prime number found as a member of an integersequence . For example, the 8th Delannoy number , 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly growing sequences. A common subclass of integersequence primes are constant primes , formed by taking a constant real number and considering prefixes of its decimal representation, omitting the decimal point. For example, the first 6 decimal digits of the constant , approximately 3.14159265, form the prime number 314159, which is therefore known as a pi prime . Similarly, a constant prime based on e mathematical constant e is called an e prime . Other examples of integersequence primes include Cullen prime &ndash a prime that appears in the sequence of Cullen numbers math a n n2 n 1 , . math Factorial prime &ndash a prime that appears in either of the sequences math a n n 1 math or math b n n 1 , . math Fermat prime &ndash a prime that appears in the sequence of Fermat numbers math a n 2 2 n 1 , . math Fibonacci prime &ndash a prime that appears in the sequence of Fibonacci number s. Lucas prime &ndash a prime that appears in the Lucas number s. Mersenne prime &ndash a prime that appears in the sequence of Mersenne numbers math a n 2 n 1 , . math Primorial prime &ndash a prime that appears in either of the sequences math a n n 1 math or math b n n 1 , . math Pythagorean prime &ndash a prime that appears in the sequence math a n 4n 1 , . math Woodall prime &ndash a prime that appears in the sequence of Woodall numbers math a n n2 n 1 , . math The On Line Encyclopedia of Integer Sequences includes many sequences corresponding to the prime subsequences of well known sequences, for example OEIS A001605 A001605 for Fibonacci number s that are prime. References MathWorld urlname IntegerSequencePrimes title IntegerSequence Primes MathWorld urlname ConstantPrimes title Constant Primes MathWorld urlname Pi Prime title Pi Prime MathWorld ... more details
have equal cardinality. See also Portal Mathematics Integer part IntegersequenceInteger lattice ...About the mathematical concept integers in computer science Integer computer science Image Latex integers.svg thumb 100px Symbol often used to denote the set of integers The integers from the Latin integer ... integers to distinguish them from the more broadly defined algebraic integer s but with rational ... of addition and multiplication , that is, the sum and product of any two integers is an integer. However ... division , since the quotient of two integers e.g. , 1 divided by 2 , need not be an integer. Although ... mathematics Closure a b is an integer a × b is an integer Associativity a b c a b ... integer can be written as a finite sum 1 1 ... 1 or &minus 1 &minus 1 ... &minus 1 . In fact, Z under ... multiplication is a commutative monoid . However not every integer has a multiplicative inverse e.g. there is no integer x such that nowrap 1 2 x 1, because the left hand side is even, while the right ... denotes the absolute value of b . The integer q is called the quotient and r is called the remainder ... domain . This implies that Z is a principal ideal domain and any positive integer can be written ... or lower bound. The ordering of Z is given by ... &minus 3 &minus 2 &minus 1 0 1 2 3 ... An integer ... or additive inverse of an integer is obtained by reversing the order of the pair math a,b b,a . , math ... in computing Main Integer computer science An integer is often a primitive datatype in computer language s. However, integer datatypes can only represent a subset of all integers, since practical ..., positive, and 0 . It is, however, certainly possible for a computer to determine whether an integer value is truly positive. Fixed length integer approximation datatypes or subsets are denoted int or Integer in several programming languages such as Algol68 , C computer language C , Java programming ... more details
. If the terms of a sequence are integer s, then the sequence is an integersequence . If the terms ... of sequences On Line Encyclopedia of Integer Sequences Permutation Recurrence relation Sequence space ...Other uses In mathematics , a sequence is an ordered list of objects or events . Like a Set mathematics ... possibly infinite is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a Discrete mathematics discrete function mathematics function . For example, C, R, Y is a sequence of letters ..., or Infinite set infinite , such as the sequence of all even and odd numbers even positive and negative numbers positive integer s 2, 4, 6,... . Finite sequences are sometimes known ... notions of sequence, but may be excluded depending on the context. Image Cauchy sequence illustration2.svg right thumb 350px An infinite sequence of real numbers in blue . This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy sequence Cauchy . It is bounded, however ... of which e.g. , exact sequence are not covered by the notations introduced below. In addition to identifying the elements of a sequence by their position, such as the 3rd element , elements may be given names for convenient referencing. For example a sequence might be written as a sub 1 sub , a sub ... definition of a finite sequence with terms in a set S is a function mathematics function from 1, 2, ..., n to S for some n 0. An infinite sequence in S is a function from 1, 2, ... to S . For example, the sequence of prime numbers 2,3,5,7,11, is the function 1 2 , 2 3 , 3 5 , 4 7 , 5 11 , . A sequence of a finite length n is also called an n tuple n tuple . Finite sequences include the empty sequence ... sequence or two way infinite sequence . An example is the bi infinite sequence of all even integers , 4, 2, 0, 2, 4, 6, 8 . Multiplicative Let A a sequence defined by a function f 1, 2, 3, ... 1, 2 ... more details
Infobox musical artist See Wikipedia WikiProject Musicians Name The Sequence Img Img capt Img size Only for images narrower than 220 pixels Landscape Background group or band Alias Origin Columbia, South Carolina Columbia , South Carolina , United States U.S. Genre Old school hip hop br Funk Years active Label Sugar Hill Records rap Sugar Hill Associated acts Spoonie Gee URL Past members Angie Stone Angie Brown Stone Angie B. br Cheryl Cook Cheryl The Pearl br Gwendolyn Chisolm Blondie The Sequence is a former female old school hip hop trio signed to the Sugar Hill Records rap Sugar Hill label in the early 1980s. The group consisted of Cheryl Cook Cheryl The Pearl , Gwendolyn Chisolm Blondie , and lead singer rapper Angie Stone Angie Brown Stone Angie B. . The group originated from Columbia, South Carolina Columbia , South Carolina as a group of high school cheerleader s. Their most notable single was Funk You Up 1979 , which was the first rap record released by a female group and the second single released by Sugar Hill Records rap Sugar Hill Records . ref name Greenberg1999 Greenberg, Steve Light, Alan ed. 1999 . The VIBE History of Hip Hop . Three Rivers Press. p. 28. ISBN 0609805037 ref Elements of Funk You Up were later used by Dr. Dre for his 1995 single Keep Their Heads Ringin ... and later a solo artist. Discography Albums Sugarhill Presents the Sequence 1980 , Sugar Hill Records rap Sugar Hill The Sequence 1982 , Sugar Hill 51 Black Albums The Sequence Party 1983 , Sugar Hill Compilations Funky Sound 1995 , P Vine The Best of the Sequence 1996 , Deep Beats Monster Jam Back ... 1982 , Sugar Hill References Reflist Hogan, Ed. Allmusic class artist id p194849 pure url yes The Sequence . Allmusic . External links http www.discogs.com artist Sequence, The Discography DEFAULTSORT Sequence, The Category African American musical groups Category African American rappers Category ... Category Musical trios Hiphop band stub no The Sequence ... more details
An integer programming problem is a mathematical Optimization mathematics optimization or Optimization mathematics feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to Linear program Integer unknowns integer linear programming , which is also known as mixed integer programming. Integer programming is NP hard . A special case, 0 1 integer linear programming, in which unknowns are binary, is one of the Karp s 21 NP complete problems . External links http mat.gsia.cmu.edu orclass integer integer.html A Tutorial on Integer Programming optimization algorithms mathapplied stub Category Operations research Category Combinatorial optimization ca Programaci lineal entera nl Geheeltallige programmering pl Programowanie ca kowitoliczbowe pt Programa o inteira ru vi Quy ho ch s nguy n ... more details
In number theory , a Gaussian integer is a complex number whose real and imaginary part are both integer ... integer s. This domain does not have a total order ing that respects arithmetic. Image Gaussian integer ... integer lattice . The Field norm norm of a Gaussian integer is the natural number defined as math ... i . If x is a Gaussian integer, the four numbers x , ix , &minus x , and &minus ix are called ... axes. The positive integer Gaussian primes are the prime numbers Congruence class congruent ... , OEIS sequence A002145 COMMENT section ref Image gauss primes 768x768.png 170px thumb Some of the Gaussian primes A Gaussian integer math a bi math is a Gaussian prime if and only if either one of a , b is zero and the other is a prime number of the form math 4n 3 math with n a nonnegative integer ... ramified prime in Z i . The integer 2 factors as math 2 1 i 1 i i 1 i 2 math as a Gaussian integer ... as following if a Gaussian integer is a Gaussian prime, then either its norm is a prime number, or its norm is a square of a prime number. This is because for any Gaussian integer math g math , notice math g mid g bar g N g math . Now math N g math is an integer, and so can be factored as a product ... have been trivial and math g math is a Gaussian prime. If math g math is a Gaussian integer whose norm ... number is within math frac sqrt 2 2 math units of a Gaussian integer. Put another way, every complex number and hence every Gaussian integer has a maximal distance of math frac sqrt 2 2 sqrt N z math units to some multiple of z, where z is any Gaussian integer this turns Z i into a Euclidean ... the integers . In a footnote he notes that the Eisenstein integer s are the natural domain for stating ... See also Hurwitz quaternion Eisenstein integer fr Entier de Dirichlet Dirichlet integer Quadratic integer Algebraic integer Proofs of Fermat s theorem on sums of two squares Proofs of quadratic reciprocity ... integers Table of Gaussian integer factorizations Notes reflist References C. F. Gauss, Theoria ... more details
italic title Taxobox name Gynoeryx integer image image width 250px image caption regnum Animal ia phylum Arthropod a classis Insect a ordo Lepidoptera familia Sphingidae genus Gynoeryx species G. integer binomial Gynoeryx integer binomial authority Viette, 1956 ref http www.cate sphingidae.org taxonomy Gynoeryx integer.html CATE Creating a Taxonomic eScience Sphingidae ref synonyms Polyptychus integer small Viette, 1956 small Polyptychus integer nigropuncta small Griveaud, 1964 small Gynoeryx integer is a moth of the Sphingidae family. It is known from Madagascar . ref http www.biodiversitylibrary.org jeanhs XXVI 20No.3 20 115 1 1967 Carcasson.pdf Revised Catalogue of the African Sphingidae Lepidoptera with Descriptions of the East African species ref References reflist Category Smerinthini Sphingidae stub vi Gynoeryx integer ... more details
In mathematics , an integer matrix is a matrix mathematics matrix whose entries are all integer s. Examples include binary matrix binary matrices , the zero matrix , the unit matrix , and the adjacency matrix adjacency matrices used in graph theory , amongst many others. Integer matrices find frequent application in combinatorics . Examples math left begin array cccc 5 & 2 & 6 & 0 4 & 7 & 3 & 8 5 & 9 & 0 & 4 3 & 1 & 0 & 3 9 & 0 & 2 & 1 end array right math and math left begin array ccc 1 & 5 & 0 0 & 9 & 2 1 & 7 & 3 end array right math are both examples of integer matrices. Properties matrix inverse Invertibility of integer matrices is in general more numerically stable than that of non integer matrices. The determinant of an integer matrix is itself an integer, thus the smallest possible magnitude of the determinant of an invertible integer matrix is one , hence where inverses exist they do not become excessively large see condition number . Theorems from matrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned matrix ill conditioned nearly zero determinant real numbers real or floating point valued matrices. The inverse of an integer matrix math M math is again an integer matrix if and only if the determinant of math M math is exactly math 1 math or math 1 math . Integer matrices of determinant math pm 1 math form the group math mathrm GL n, mathbf Z math , which has far reaching applications in arithmetic and geometry. For math n 2 math , it is closely related to the modular group . The intersection of the integer ... polynomial of an integer matrix has integer coefficients. Since the eigenvalue s of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are algebraic integers . In dimension ... integers. Integer matrices are sometimes called integral matrices , although this use is discouraged. See also Unimodular matrix External links http mathworld.wolfram.com IntegerMatrix.html Integer ... more details
summary in Repeating decimal Mergeto cyclic number discuss Talk Cyclic number Cyclic permutation of integer ... fractions , or directly. General For any integer coprime to 10, its reciprocal is a repeating decimal ... integer and 10 sup m sup &minus 1 is constant. Expressed as a formula, math gcd left N,10 m 1 right gcd left N c,10 m 1 right , math where N is an m digit integer and N sub c sub is any ... 509157, 999999 If N is an m digit integer, the number N sub c sub , obtained by shifting N to the left ... terms , have the same denominator. This is true for cyclic permutations of any integer. Fraction method Integral multiplier An integral multiplier refers to the multiplier n being an integer An integer X shift right cyclically by k positions when it is multiplied by an integer n . X is then the repeating ... , or a factor of F sub 0 sub excluding any values of F which are not more than n . An integer X shift left cyclically by k positions when it is multiplied by an integer n . X is then the repeating ... provided that the integer i satisfies the condition frac n j F 1. Most often it is convenient ... an integer j such that frac j F frac 1 10 , i.e. j frac F 10 . There is no solution when n F . Fractional multiplier An integer X shift left cyclically by k positions when it is multiplied by a fraction ... of X , i.e. j X are again solutions but the condition to be satisfied for integer j is that frac ..., select an integer j such that frac j s F frac 1 10 , i.e. j frac F 10 s . Again if frac ... of an integer t is often abbreviated math operatorname gcd left 10 infty,t right . math If the numbers ... by multiplication. E.g., The integer 142857, corresponding to remainder 1, permutes to 428571 when multiplied by 3, the corresponding remainder of the latter. The integer 142857, corresponding .... The integer 857142, corresponding to remainder 6, permutes to 571428 when multiplied by frac 5 6 ... can be applied to any integer to cyclic shift shift cyclically right or left by any given number ... more details
orphan date April 2010 Taxobox image image caption regnum Animal ia phylum Arthropod a classis Insect a ordo Fly Diptera zoosectio Schizophora zoosubsectio Acalyptratae superfamilia Tephritoidea familia Ulidiidae genus Tetanops species T. integer binomial Tetanops integer subdivision ranks Subspecies subdivision Tetanops integer is a species of ulidiid or picture winged fly in the genus Tetanops of the family Tephritidae . ref http www.itis.gov servlet SingleRpt SingleRpt ref ref http nomen.at Tetanops integer ref ref http data.gbif.org species browse taxon 13143053 ref References Reflist Category Tetanops Ulidiidae stub ... more details
unreferenced date August 2007 In mathematics , a half integer is a number of the form math n 1 2 math , where math n math is an integer . For example, 4½ , 7 2, &minus 13 2, 8.5 are all half integers. Note that a half of an integer is not always a half integer half of an even integer is an integer but not a half integer. The half integers are precisely those numbers that are half of an odd integer . The Set mathematics set of all half integers is often denoted math mathbb Z 1 over 2 . math Uses Half integers occur frequently enough in mathematical contexts that a special term for them is convenient. For example, the densest lattice packing of unit sphere s in four dimensions places a sphere at every point whose coordinates are either all integers or all half integers this packing is closely related to the Hurwitz integer s, which are quaternion s whose real coefficients are either all integers or all half integers. Moreover, the Pauli exclusion principle results from definition of fermion s as particles which have spin physics spin s that are half integers. The energy level s of the quantum harmonic oscillator occur at half integers and thus its lowest energy is not zero. Also, the factorial and gamma function s, while not defined for negative number negative and non positive integers respectively, are defined for all half integers as rational multiples of the square root of pi . Category Rational numbers Category Elementary number theory Category Parity de Halbzahlig et Poolarv fr Demi entier ko it Semidispari hu F leg sz sz mok ja ru fi Puoliluku zh ... more details
Unreferenced date December 2009 In mathematics , the n dimensional integer lattice or cubic lattice , denoted Z sup n sup , is the lattice group lattice in the Euclidean space R sup n sup whose lattice points are n tuples n tuples of integer s. The two dimensional integer lattice is also called the square lattice , or grid lattice. Z sup n sup is the simplest example of a root lattice . The integer lattice is an odd unimodular lattice . Automorphism group The automorphism group or group of congruence relation congruence s of the integer lattice consists of all permutation s and sign changes of the coordinates, and is of order 2 sup n sup n nowiki nowiki . As a matrix group it is given by the set of all n × n signed permutation matrices . This group is isomorphic to the semidirect product math mathbb Z 2 n rtimes S n math where the symmetric group S sub n sub acts on Z sub 2 sub sup n sup by permutation this is a classic example of a wreath product . For the square lattice, this is the group of the square, or the dihedral group of order 8 for the three dimensional cubic lattice, we get the group of the cube, or octahedral group , of order 48. Diophantine geometry In the study of Diophantine geometry , the square lattice of points with integer coordinates is often referred to as the Diophantine plane . In mathematical terms, the Diophantine plane is the Cartesian product math scriptstyle mathbb Z times mathbb Z math of the ring of all integers math scriptstyle mathbb Z math . The study of Erd s Diophantine graph Diophantine figures focusses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer. Coarse geometry In coarse structure coarse geometry , the integer lattice is coarsely equivalent to Euclidean space . See also Regular grid DEFAULTSORT Integer Lattice Category Euclidean geometry Category Lattice points Category Diophantine geometry ... more details
Italic title Taxobox name Neomysis integer regnum Animal ia phylum Arthropod a subphylum Crustacean Crustacea classis Malacostraca ordo Mysida familia Mysidae genus Neomysis species N. integer binomial Neomysis integer binomial authority William Elford Leach Leach , 1815 ref ITIS id 90063 taxon Neomysis integer Leach, 1815 accessdate November 17, 2010 ref Neomysis integer is a species of Mysidacea opossum shrimp found in shallow marine Headlands and bays bays of Europe , with a transparent greenish or brown body and a large cephalothorax . ref name Scheffel cite book author Christoph Needon, Johannes Petermann, Peter Scheffel & Bernd Scheiba title Naturf hrer Pflanzen und Tiere year 1989 ISBN 3332002562 pages 279 publisher Urania Verlag location Leipzig language German ref These opossum shrimp are frequently found in large numbersand filter small food particles out of the water, occasionally consumimg larger prey. Opossum shrimps are themselves often consumed by fish es. ref name Scheffel The females have a brood pouch Peracarida brood pouch in which eggs and larva e develop. ref name Scheffel References reflist Category Mysida Category Animals described in 1815 crustacean stub ... more details
about the ring of complex numbers integral over math the general notion of algebraic integer Integrality Distinguish algebraic element In number theory , an algebraic integer is a complex number that is a root ... K . Each algebraic integer belongs to the ring of integers of some number field. A number x is an algebraic integer if and only if the ring math x is Finitely generated group finitely generated ... definitions of an algebraic integer. Let K be a number field i.e., a finite extension of math mathbb ... element theorem . math alpha in K math is an algebraic integer if there exists a monic polynomial ... integer if the minimal monic polynomial of math alpha math over math mathbb Q math is in math mathbb Z x math . math alpha in K math is an algebraic integer if math mathbb Z alpha math is a finitely generated math mathbb Z math module. math alpha in K math is an algebraic integer if there exists .... In particular, an algebraic integer is an integral element of a finite extension math K mathbb Q ... integer unless b divides a . Note that the leading coefficient of the polynomial bx &minus a is the integer b . As another special case, the square root n of a non negative integer n is an algebraic integer, and so is irrational unless n is a square number perfect square . If d is a square free integer then the extension K Q overline d is a quadratic field extension ... 1 mod 4 the element 1 overline d 2 is also an algebraic integer. It satisfies ... the constant term 1 &minus d 4 is an integer. The full ring of integers is generated by overline ... , then the ring of integers of the cyclotomic field Q is precisely Z . If is an algebraic integer then math beta sqrt n alpha math is another algebraic integer. A polynomial for is obtained ... which has integer coefficients but is not monic, and P is irreducible polynomial irreducible over Q ... is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally ... more details
refimprove article date September 2007 Integer BASIC , written by Steve Wozniak , was the BASIC interpreter ... took about 30 seconds to load ... ref Thousands of programs were written in Integer BASIC. Little ... 2007 09 16 quote The nowiki Integer nowiki BASIC, which we shipped with the first Apple II .... ref Relationship to Applesoft BASIC The most frequently cited flaw of Integer BASIC was, as one might expect from the name, that its variables were all 16 bit integer computer science integers and it was very .... Integer BASIC was relegated to a file on the system floppy disk that Apple II Plus users could ... more programs than Integer BASIC. However, loading the Integer BASIC language from floppy disk is possible even on the latest models of the Apple II line, should the need or desire arise. Integer BASIC .... Applesoft BASIC uses floating point for all numerical operations, even in cases where integer would ... being performed by Integer BASIC at entry time, as well as numbers being converted to binary file ... of iteration Computing iterations . This method is superfluous in Integer BASIC. The Mini Assembler and other auxiliary firmware The Integer BASIC ROMs also included a Mini Assembler that let programmers ... input output I O routines, and a few seldom used floating point math routines which existed in the Integer ... the Integer BASIC ROMs to the Apple II Plus ROMs, in order to accommodate the larger size of the Applesoft ... of clearing the current BASIC program from memory to switch between Applesoft BASIC and Integer BASIC by typing either INT to enter Integer BASIC or FP to enter Applesoft BASIC provided, of course ... or external ROMS. The command line prompt for Integer BASIC was a right facing arrowhead greater than ... BASIC they were using. Editing The editing method for Integer BASIC actually part of the system monitor ... then A and move just one space. If Integer BASIC is loaded from a DOS 3.3 disk into an Apple II ... BASIC External links http www.landsnail.com a2ref2.htm INTEGER BASIC Reference &ndash From Landsnail.com ... more details
Image Odometer rollover.jpg thumb 250px Odometer rollover, a mechanical form of integer overflow. All digits are set to the maximum 9, and adding 1 causes a cascade of carry over additions setting all digits to 0 but there is no higher digit to change to a 1, so the counter resets to zero. In computer programming , an integer overflow occurs when an arithmetic operation attempts to create a numeric ... 1 to the largest value that can be represented constitutes an integer overflow. The most common result ... language C programming language , signed integer overflow causes undefined behavior , while unsigned integer overflow causes the number to be reduced modular arithmetic modulo a power of two , meaning ... number. In counting, one just starts over again from the bottom. Example 16 bit signed integer 30000 30000 5536. Diagram that illustrates wrapping behavior of integer representation. In computer graphics ... has a signed integer type an overflow can cause its value to wrap and become negative, violating ... too large for an integer seamlessly becomes a long. ref http www.python.org doc 1.4 ref ref5.html ... 237 ref Techniques for mitigating integer overflow problems List of techniques and methods that might be used to mitigate the consequences of integer overflow The effects of integer based attacks for C C and how to defend against them by using subtyping in http www.cs.cmu.edu dbrumley pubs integer ndss 07.pdf Efficient and Accurate Detection of Integer based Attacks . CERT As if Infinitely Ranged AIR integer model a largely automated mechanism for eliminating integer overflow and integer truncation http www.cert.org archive pdf 09tn023.pdf As if Infinitely Ranged Integer Model See also Arithmetic ... 60&id 10 article Phrack 60, Basic Integer Overflows http www.phrack.org issues.html?issue 60&id 9 article Phrack 60, Big Loop Integer Protection http thetaeng.com TimerWrap.htm How to implement efficiently in C http projects.webappsec.org Integer Overflows WASC Threat Classification Integer Overflows ... more details
Infobox company company logo Image Intger Group Logo.jpg 170px name The Integer Group type Advertising Agency location city Denver, CO location country United States industry Promotional, Retail, and Shopper Marketing parent company Subsidiary of Omnicom Group homepage http www.integer.com http www.integer.com The Integer Group is one of the world s largest promotional, retail, and shopper marketing agencies. The Integer Group is a part of the Omnicom Group Inc. and a member of TBWA Worldwide . It provides services in the following disciplines promotional marketing, retail & shopper marketing, account leadership, project management, creative, media, http www.integervelocidad.com Velocidad Hispanic Marketing , digital, art production services, public relations, and insight & strategy. ref http www.integer.com portfolio capabilities ref In 2001, Integer officially became part of Omnicom . First as part of DAS and then, five years later, with TBWA as part of its global agency group. Today, Integer has expanded to form its own network of agencies including five core US offices, over 20 field offices, and 9 global offices. Offices The Integer Group headquarters are in Denver, Colorado . U.S. Offices are located in Chicago, Dallas, Des Moines, and New York. International offices include Mexico City, Dubai, Hamburg, Johannesburg, Milan, Paris, Sao Paulo, Shanghai, Singapore, and Tokyo. ref http www.integer.com contact ref ref http biz.yahoo.com ic 116 116781.html ref ref http www.redbooks.com Subscriber Profile.aspx?ent 353314 ref Abridged client list ref http www.integer.com portfolio clients ref 7 Eleven Allsteel AT&T BancVue Benjamin Moore & Company BFGoodrich Tires BG Products Bimbo Bakeries Black & Decker Carrefour Columbia Flooring Dal Tile Danone Electrolux FedEx Office Frigidaire GlaxoSmithKline Holmes Murphy & Associates Illinois Lottery Innovative Captive Strategies ... Integer s Web site http www.integer.com Shopper Culture Blog http www.shopperculture.com Velocidad ... more details
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers ... triangle can be integrally rescaled can have all sides multiplied by the same integer, namely a common multiple of their denominators to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. Note however, that other definitions ... triangle has a right angle and three integer sides, which are known as a Pythagorean triple ... of them is even with m n . Pythagorean triangles with integer altitude from the hypotenuse All primitive Pythagorean triangles with legs a b , hypotenuse c , and integer altitude math ... tfrac 1 b 2 tfrac 1 d 2 math , are generated by ref Voles, Roger, Integer solutions of math a 2 b ... y and hypotenuse z can generate another Pythagorean triangle, this one with integer altitude d ... , is a triangle with integer sides and integer area. All Heronian triangles can be generated ref ... with sides in arithmetic progression A triangle with integer sides and integer area has sides ... m 2 3n 2, math math 2mn math , and math m 2 3n 2. math Integer triangles with a 60 angle All primitive integer triangles with a 60 angle can be generated by ref Gilder, J., Integer sided triangles ... 3 . See also ref name Burn Burn, Bob, Triangles with a 60 angle and sides of integer length, Mathematical Gazette 87, March 2003, 148&ndash 153. ref ref name Read Read, Emrys, On integer sided triangles containing angles of 120 or 60 , Mathematical Gazette 90, July 2006, 299 305. ref Integer triangles with a 120 angle All primitive integer triangles with a 120 angle can be generated by ref Selkirk, K., Integer sided triangles with an angle of 120 , Mathematical Gazette 67, December 1983, 251 ... mod 3 . See also. ref name Burn ref name Read Integer triangles with one angle equal to twice ... 2 n . Note that all triangles with B 2 A whether integer or not have ref Willson ... more details
Unreferenced date December 2009 In computer programming , a short integer is a data type that can represent a whole number which may take less storage, while having a smaller range, compared with a standard integer computer science integer on the same machine. A variable programming variable defined as a short integer in one programming language may be different in size to a similarly defined variable in another. In some languages this size is fixed across platforms, while in others it is machine dependent. In some languages this datatype does not exist at all. In C programming language C , it is denoted by code short code . It is required to be at least 16 bits, and is often smaller than a standard integer, but this is not required. ref name c99 ref name drdobbsinteger A conforming program can assume that it can safely store values between &minus 2 sup 15 sup &minus 1 and 2 sup 15 sup &minus 1, but it may not assume that the range isn t larger. In Java programming language Java , a code short code is always a 16 bit integer. In the Windows API , the datatype code SHORT code is defined as a 16 bit signed integer on all machines. ref name agnerfog Common sizes class wikitable Programming language Platform computing Platform s Data type name Signedness Storage in bytes Minimum value Maximum value rowspan 2 C programming language C and C rowspan 2 common implementations code short code signed 2 &minus 32,768 32,767 code unsigned short code unsigned 2 0 65,535 rowspan 2 C Sharp programming language C rowspan 2 .NET Framework .NET CLR CTS code short code signed 2 &minus 32,768 32,767 tt ushort tt unsigned 2 0 65,535 Java programming language Java Java platform code short code signed 2 &minus 32,768 32,767 References Reflist refs ref name drdobbsinteger cite web url http www.drdobbs.com 184401323 title The New C Integers in C99, Part 1 date 2000 12 01 accessdate 2010 09 04 last ... See also Long integer DEFAULTSORT Short Integer Category Computer arithmetic Category Data types fa ... more details
In mathematics , a natural number n is a Blum integer if n p q is a semiprime for which p and q are distinct prime number s congruent to 3 modular arithmetic mod 4. ref name HurdBlum Joe Hurd, Blum Integers 1997 , retrieved 17 Jan, 2011 from http www.gilith.com research talks cambridge1997.pdf ref That is, p and q must be of the form 4 t 3, for some integer t . Primes of this form are referred to as Blum primes. ref name GoldBell Shafi Goldwasser Goldwasser, S. and Mihir Bellare Bellare, M. http cseweb.ucsd.edu mihir papers gb.html Lecture Notes on Cryptography . Summer course on cryptography, MIT, 1996 2001 ref This means that the factors of a Blum integer are Gaussian integer Gaussian primes with no imaginary part. The first few Blum integers are 21 number 21 , 33 number 33 , 57 number 57 , 69 number 69 , 77 number 77 , 93 number 93 , 129 number 129 , 133 number 133 , 141 number 141 , 161 number 161 , 177 number 177 , ... OEIS id A016105 Blum integers were named for computer scientist Manuel Blum . Properties Given n p q a Blum integer, Q sub n sub the set of all quadratic residue s modulo n, and a Q sub n sub . Then ref name GoldBell ref a has precisely four square roots modulo n , exactly one of which is also in Q sub n sub The unique square root of a in Q sub n sub is called the principal square root of a modulo n The function f Q sub n sub Q sub n sub defined by f x x sup 2 sup mod n is a permutation. The inverse function of f is f sup 1 sup x x sup p 1 q 1 4 8 sup mod n . ref name hac Alfred Menezes A.J. Menezes , P.C. van Oorschot, and Scott Vanstone S.A. Vanstone , http www.cacr.math.uwaterloo.ca hac Handbook of Applied Cryptography ISBN 0 8493 8523 7. ref For every Blum integer n , 1 has a Jacobi symbol mod n of 1, although 1 is not a quadratic residue of n math left frac 1 n right left frac 1 p right left frac 1 q right 1 2 1 math History Before modern factoring ... selected primes. References reflist Category Integer sequences numtheory stub fr Entier de Blum ... more details
a Dedekind domain, a quadratic integer ring is a UFD if and only if it is a principal ideal domain i.e., its class number is one. However, there are quadratic integer rings that are principal ideal ... ideal class group s of quadratic integer rings, but many theoretical questions about their structure ... more details
In number theory , the radical of a positive number positive integer n is defined as the product of the prime number s dividing n math displaystyle mathrm rad n prod scriptstyle p mid n atop p text prime p. , math Examples Radical numbers for the first few positive integers are 1 number 1 , 2 number 2 , 3 number 3 , 2, 5 number 5 , 6 number 6 , 7 number 7 , 2, 3, 10 number 10 , ... OEIS A007947 . For example, math 504 2 3 cdot3 2 cdot7 math and therefore math mathrm rad 504 2 cdot3 cdot7 42. , math Properties The function math mathrm rad math is multiplicative function multiplicative . The radical of any integer n is the largest square free integer square free divisor of n , and so also described as the square free kernel of n . One of the most striking applications of the notion of radical occurs in the abc conjecture , which states that, for any 0, there exists a finite K sub sub such that, for all triples of coprime positive integers a , b , and c satisfying a b c , math c K varepsilon , operatorname rad abc 1 varepsilon . math Furthermore, it can be shown that the nilpotent elements of math mathbb Z n mathbb Z math are all of the multiples of rad n . References cite book last Guy first Richard K. authorlink Richard K. Guy title Unsolved Problems in Number Theory publisher Springer Verlag date 2004 page 102 isbn 0 387 20860 7 Category Number theory Category Multiplicative functions numtheory stub de Zahlentheoretische Funktion Multiplikative Funktionen es Radical de un entero fr Radical d un entier he fi Kokonaisluvun radikaali ... more details
In computer science , a long integer is a data type that can represent a whole number whose range computer science range is greater than or equal to that of a standard integer computer science integer on the same machine. A variable defined as a long integer in one programming language may be different in size to a similarly defined variable in another. In some languages this size is fixed across platforms, whilst in others it is machine dependent. In some languages this data type does not exist at all. In practice it is usual for a long integer to require double the storage capacity of a standard integer, although this is not always the case. C C language dialects In later C99 version of the C programming language C programming language , a CODE long long CODE type is supported that doubles the minimum capacity of the standard CODE long CODE to 64 bits. This type is not supported by compilers that require C code to be C International Organization for Standardization ISO compliant, because the code long long code type does not currently exist in C it is defined in the new, yet unfinished, C standard, C 0x . For an ANSI ISO compliant compiler the minimum requirements for the specified ranges, that is &minus 2 sup 31 sup &minus 1 to 2 sup 31 sup &minus 1 for signed and 0 to 2 sup 32 sup &minus 1 for unsigned, must be fulfilled however, extending this range is permitted. ref cite web url http www.ericgiguere.com articles ansi c summary.html title The ANSI Standard A Summary for the C Programmer first Eric last Giguere date December 18, 1987 accessdate 2010 09 04 ref ref cite web ... aa164754 office.10 .aspx title The Integer, Long, and Byte Data Types VBA accessdate 2006 12 19 ... Meyers first Randy publisher drdobbs.com ref See also computer numbering formats integer computer science integer short integer External links http www.netrino.com Articles FixedWidthIntegers Portable Fixed Width Integers in C Category Computer arithmetic Category Data types it Long integer ... more details
Integer Circuit computer theory circuit s is a mathematical model used in studying computational complexity theory . It is a special case of circuit computer theory circuit , the object is a labeled directed acyclic graph the nodes of which evaluate to sets of integers, the leaves are finite sets, and the gates are set operations or arithmetic operations. As an algorithm ic problem, the possible question are to find if a given integer is an element is in the output node or if two circuits compute the same set. The decidability is still an open question, but there are results on restriction of thoses circuits. Finding answers to some questions about this model could serve as a proof to many important mathematical conjectures, like Goldbach s conjecture . It is a natural extension of the circuits over sets of natural numbers when the considered set contains also negative integers, the definitions, which does not change, will not be repeated on this page. Only the differences will be mentioned. Complexity results Membership problem The membership problem ask if, given an element math n math and a circuit, if math n math is in the output gate of the circuit. When the class of authorized gate is restricted, the membership problem lay inside well kwown complexity classes ref Citation author Stephen Travers title The complexity of membership problems for circuits over sets of integers journal Theoretical Computer Science volume 369 edition what is called number in bibtex issue 1 year 2006 issn 0304 3975 pages 211&ndash 229 publisher Elsevier Science Publishers Ltd place Essex, UK url http portal.acm.org citation.cfm?id 1238761 ref . class wikitable style text align center width 80 Complexity scope col O scope col MC math mathbb Z math O scope col MF math mathbb Z math O scope row , ,&minus , , NEXPTIME hard PSPACE hard scope row , , , NEXPTIME complete NP complete scope row , , NEXPTIME complete NP complete scope row , , P complexity P hard, in co NP L complexity L hard, in LOGCFL ... more details
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