In number theory , the classes of Lucaspseudoprime and Fibonacci pseudoprime comprise sequences of composite number composite integers that passes certain tests that all Prime number prime s pass in this case, criteria relative to some Lucas sequence . Definition Given two integer parameters P and Q which satisfy math D P 2 4Q neq 0 , math the Lucas sequences of the first and second kind, U sub n sub P , Q and V sub n sub P , Q respectively, are defined by the recurrence relation s math U 0 P,Q ... ne 0 math , then p is a factor of U sub p sub . Lucas pseudoprimes A Lucaspseudoprime ref cite journal coauthors Samuel S. Wagstaff, Jr. title Lucas Pseudoprimes journal Mathematics of Computation ... Lucaspseudoprime is an odd composite number n with n,D 1, and n 2 sup r sup s with s odd, satisfying ... r . A strong Lucaspseudoprime is also a Lucaspseudoprime. An extra strong Lucaspseudoprime is a strong Lucaspseudoprime for a set of parameters P , Q where Q 1, satisfying one of slightly modified ... j sup s sub for some j < r . An extra strong Lucaspseudoprime is also a strong Lucaspseudoprime. By combining a Lucaspseudoprime test with a Fermat primality test , say, to base 2, one can obtain ... FibonacciPseudoprime title Fibonacci Pseudoprime MathWorld urlname LucasPseudoprime title LucasPseudoprime MathWorld urlname StrongLucasPseudoprime title Strong LucasPseudoprime MathWorld urlname ExtraStrongLucasPseudoprime title Extra Strong LucasPseudoprime Category Fibonacci numbers Category ... 1. In the specific case of the Fibonacci sequence , where P 1, Q 1 and D 5, the first Lucas pseudoprimes ... by Crandall and Riesel. Fibonacci pseudoprimes A Fibonacci pseudoprime is a composite number n for which V sub n sub is congruent to P modulo n when Q 1. A strong Fibonacci pseudoprime may be defined as a composite number which is a Fibonacci pseudoprime for all P . It follows see M ller and Oswald ... of a strong Fibonacci pseudoprime is 443372888629441, which has factors 17, 31, 41, 43, 89, 97 ... more details
In mathematics , in particular number theory , an odd number odd composite number N is a Somer Lucas d pseudoprime with given d 1 if there exists a nondegenerate Lucas sequence math U P,Q math with the discriminant math D P 2 4Q, math such that math gcd N,D 1 math and the rank appearance of N in the sequence U P , Q is math frac 1 d left N left frac D N right right , math where math left frac D N right math is the Jacobi symbol . References MathWorld urlname Somer LucasPseudoprime title Somer LucasPseudoprime cite book author Ribenboim, P. chapter 2.X.D Somer Lucas Pseudoprimes chapter url http books.google.com books?id 72eg8bFw40kC&pg PA131 title The New Book of Prime Number Records edition 3rd ed. place New York publisher Springer Verlag pages 131 132 year 1996 DEFAULTSORT Somer LucasPseudoprime Category Pseudoprimes ... more details
A pseudoprime is a probable prime an integer that shares a property common to all prime number s that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Pseudoprimes are of primary importance in public key cryptography , which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost 10 million to factor a number with 144 digits, and 100 billion to factor a 200 digit number. ref cite book first Calvin C. last Clawson title Mathematical Mysteries The Beauty and Magic of Numbers location Cambridge publisher Perseus year 1996 page 195 isbn 0738202592 ref However, finding and factoring the proper prime numbers for this use is correspondingly expensive, so various probabilistic primality test s are used to find primes amongst large numbers, some of which in rare cases incorrectly identify composite number s as primes. On the other hand, deterministic primality tests, such as the AKS primality test , work well on all numbers, so there are no pseudoprimes with respect to them. Fermat pseudoprimes main Fermat pseudoprime Fermat s little theorem states that if p is prime and a is coprime to p , then a sup p 1 sup 1 is Divisor divisible by p . If a composite integer x is coprime to an integer a 1 and x divides a sup x 1 sup 1, then x is called a Fermat pseudoprime to base a . Some sources use variations of this definition, for example to only allow odd numbers to be pseudoprimes. ref MathWorld title Fermat Pseudoprime urlname FermatPseudoprime ref An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number . Classes div col colwidth 25em Fermat pseudoprime Frobenius pseudoprime Euler pseudoprime Euler Jacobi pseudoprime Extra strong Lucaspseudoprime Fibonacci pseudoprimeLucaspseudoprime Perrin pseudoprime Somer Lucaspseudoprime Strong Lucaspseudoprime Strong pseudoprime div ... more details
In number theory , a Frobenius pseudoprime is a composite number which passes a three step probable prime test set out by Jon Grantham in section 3 of his paper Frobenius pseudoprimes . ref Jon Grantham. http www.pseudoprime.com pseudo1.pdf Frobenius pseudoprimes . Mathematics of Computation , 70 234 873 891. 2001. ref Although a single round of Frobenius is slower than a single round of most standard tests, it has the advantage of a much smaller worst case per round error bound of 1 7710, which would require 7 rounds to achieve with the Miller Rabin primality test according to best known bounds. Strong Frobenius pseudoprimes A strong Frobenius pseudoprime is a pseudoprime which obeys an additional restriction beyond that required for a Frobenius pseudoprime. ref MathWorld title Strong Frobenius pseudoprime urlname StrongFrobeniusPseudoprime ref See also Pseudoprime Ferdinand Georg Frobenius References references External links http www.mathpages.com home kmath003 kmath003.htm Symmetric Pseudoprimes at MathPages Category Pseudoprimes Category Article Feedback 5 Numtheory stub ... more details
In number theory , a strong pseudoprime is a composite number that passes a primality test. All primes pass this test, but a small fraction of composites pass as well, making them pseudoprime false primes . Unlike the Fermat pseudoprime s, for which there exist numbers that are pseudoprimes to all coprime bases the Carmichael numbers , there are no composites that are strong pseudoprimes to all bases. Formal definition Formally, a composite number n d 2 sup s sup 1 with d being odd is called a strong pseudoprime to a relatively prime base a when one of the following conditions hold math a d equiv 1 mod n math math a d cdot 2 r equiv 1 mod n quad mbox for some 0 leq r leq s 1 math The definition of a strong pseudoprime depends on the base used different bases have different strong pseudoprimes. The definition is trivially met if math var a var 1 mod var n var so these base choices are often excluded. Richard K. Guy Guy mistakenly gives a definition with only the first condition, which is not satisfied by all primes. ref Richard K. Guy Guy , Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes. A12 in Unsolved Problems in Number Theory , 2nd ed. New York Springer Verlag, pp. 27 30, 1994. ref Properties of strong pseudoprimes A strong pseudoprime to base a is always an Euler pseudoprime ref name PSW Carl Pomerance Pomerance , John Selfridge Selfridge and Samuel S. Wagstaff Jr. Wagstaff , The pseudoprimes to 25 10 sup 9 sup . Mathematics of Computation , 35 151 pp. 1003 1026, 1980. ref and a Fermat pseudoprime to that base, but not all Euler and Fermat pseudoprimes are strong pseudoprimes. Carmichael number s may be strong pseudoprimes to some bases for example, 561 is a strong pseudoprime to base 50 but not to all bases. A composite number n is a strong pseudoprime to at most one quarter of all bases below n ref Louis Monier Monier , Evaluation and Comparison of Two Efficient ... pseudoprime to that base is less than 1 4, forming the basis of the widely used Miller Rabin primality ... more details
In arithmetic , an Odd number odd composite number composite integer n is called an Euler pseudoprime to base a , if a and n are coprime , and math a n 1 2 equiv pm 1 pmod n math where mod refers to the modular arithmetic modulo operation . The motivation for this definition is the fact that all prime number s p satisfy the above equation which can be deduced from Fermat s little theorem . Fermat s theorem asserts that if p is prime, and coprime to a , then a sup p &minus 1 sup 1 mod p . Suppose that p 2 is prime, then p can be expressed as 2 q 1 where q is an integer. Thus a sup 2 q 1 &minus 1 sup 1 mod p which means that a sup 2 q sup &minus 1 0 mod p . This can be factored as a sup q sup &minus 1 a sup q sup 1 0 mod p which is equivalent to a sup p &minus 1 2 sup 1 mod p . The equation can be tested rather quickly, which can be used for probabilistic prime testing primality testing . These tests are twice as strong as tests based on Fermat s little theorem. Every Euler pseudoprime is also a Fermat pseudoprime . It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes , numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael number s, and the smallest absolute Euler pseudoprime is 1729 number 1729 7× 13× 19. The stronger condition that a sup n &minus 1 2 sup a n mod n , where a , n 1 and a n is the Jacobi symbol , is sometimes used for a definition of an Euler pseudoprime. A discussion of numbers of this form can be found at Euler&ndash Jacobi pseudoprime . See also Probable prime References N. Koblitz, A Course in Number Theory and Cryptography , Springer Verlag, 1987. H. Riesel, Prime ... Pseudoprime Category Pseudoprimes de Eulersche Pseudoprimzahl fr Nombre pseudopremier d Euler it Pseudoprimo ... more details
In number theory , the Fermat pseudoprimes make up the most important class of pseudoprime s that come from Fermat s little theorem . Definition Fermat s little theorem states that if p is prime and a is coprime to p , then a sup p 1 sup 1 is Divisor divisible by p . If a composite integer x is coprime to an integer a 1 and x divides a sup x 1 sup 1, then x is called a Fermat pseudoprime to base a . In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes Fermat primality test for the base a . ref name desmedt 10 23 cite book author Desmedt, Yvo chapter Encryption Schemes editors Mikhail Atallah Atallah, Mikhail J. & Blanton, Marina title Algorithms and theory of computation handbook Special topics and techniques publisher CRC Press year 2010 isbn 978 1 58488 820 8 pages 10 23 url http books.google.com books?id SbPpg 4ZRGsC&pg SA10 PA23 ref The smallest base 2 Fermat pseudoprime is 341. It is not a prime, since it equals 11 31, but it satisfies Fermat s little theorem 2 sup 340 sup 1 mod 341 and thus passes Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Poulet numbers , Sarrus numbers , or Fermatians OEIS id A001567 . An integer x that is a Fermat pseudoprime for all values of a that are coprime ... title Fermat Pseudoprime urlname FermatPseudoprime ref Every odd number q satisfies math a q 1 equiv 1 pmod q math for math a q 1 math . This trivial case is excluded in the definition of a Fermat pseudoprime ... year 2001 page 132 chapter Theorem 3.4.2 ref A composite number q is a Fermat pseudoprime to a base ... numbers which are not super Poulet Numbers. Smallest Fermat pseudoprimes The smallest pseudoprime ... Jacobi pseudoprime Another approach is to use more refined notions of pseudoprimality, e.g. strong pseudoprime s or Euler Jacobi pseudoprime s, for which there are no analogues of Carmichael number ... found is not a prime number but a pseudoprime, it is possible to use the much faster ... more details
wiktionarypar Lucas TOCright Lucas is the Latin language Latin form of the Greek language Greek name Loukas disambiguation Loukas , and is a common name in many modern languages. Lucas may refer to several things. People bearing the surname Lucas See Lucas surname People bearing the given name Lucas Real Lucas Barrios , Paraguayan footballer currently playing for Borussia Dortmund Lucas Biglia , Argentine footballer currently playing for RSC Anderlecht Lucas Bolsius , Dutch historian and politician Lucas Bryant , actor Lucas Cranach the Elder , painter Lucas Cranach the Younger , painter Lucas Cruikshank , teenage internet celebrity Lucas Grabeel , actor Lucas Leiva , Brazilian footballer currently playing for Liverpool Lucas Matthysse , Argentine boxer Lucas Neill , Australian footballer currently playing for Galatasaray Lucas Papademos , Greek bank official and prime minister Lucas P rez Mart nez , Spain footballer currently playing for FC Karpaty Lviv Lucas Rios Marques , Brazilian footballer currently playing for Figueirense Lucas musician Lucas Secon , Danish American rapper Lucas Severino , Brazilian footballer currently playing for FC Tokyo Lucas Vita , Brazilian water polo player Lucas Zen , Brazilian footballer currently playing for Botafogo Jefferson Lucas Azevedo dos Santos , Brazilian footballer Lucas da Silva Lucas , Brazilian footballer currently playing for FC Kaiserslautern Lucas Rodrigues Moura da Silva Lucas , Brazilian footballer currently playing for S o Paulo F.C. Fictional Lucas Mother 3 , main character of the 2006 video game Mother 3 Lucas, male protonagist of the games Pok mon Diamond and Pearl Pok mon Diamond and Pearl Lucas Bly, main character of the 1986 film, Lucas film LucasLucas Fitzgerald , character in the Network Ten soap opera Neighbours Lucas Hay , character in Channel 4 soap opera Hollyoaks Characters in FlashForward television show Lucas Hellinger , character in the television series, FlashForward Lucas Holden , character ... more details
Infobox Company company name S. Lucas Ltd company logo File Lucas Logos small.jpg x250px File Officialpartnermclaren.png x217px foundation 1971 by Sam Lucas location Wrotham, Kent , United Kingdom industry Spray Painting, Coatings and Fit Out services key people Danny Lucas, Chairman and Managing Director homepage http www.lucasuk.com www.lucasuk.com S. Lucas Group is a specialist applied finishes, spray & painting contractor and a fit out specialist to the construction industry. It is based in Wrotham, Kent. History S. Lucas was founded by Sam Lucas in 1971 and its sister company, http www.lucasfitout.com Lucas Fit Out , was established in 2008. Sam s eldest son, Danny Lucas, joined the company in 1982 and became the Managing Director in 1997. In late 2010 he acquired his father s shareholding and became Chairman on his father s retirement from the day to day business ref http www.metacf.co.uk transactions 98 Management Buyout Meta Corporate Finance ref . The Commercial Director Ken Mulholland and Danny s brother Adam, both of whom have worked for the company for 10 years, also became shareholders . In March 2010, S. Lucas won the Finishing and Fit Out Category in the Specialist Awards 10 organised by the magazine Construction News and was shortlisted for the same award in 2011 ref http www.cnplus.co.uk home awards specialists awards shortlist announced 8609867.article Awards Specialist ... S. Lucas provides a range of decorative and protective coatings http www.lucasuk.com services.cfm ... paint effects. Its sister company, Lucas Fit Out, provides a commercial buildings finishing ... being built at London Bridge ref http www.theconstructionindex.co.uk press view 28 10 2010 s lucas ... painting, decorating and small building service. Its sister company Lucas Fit Out provides a commercial ... at Heathrow and the refurbishment of the Savoy Hotel in London ref http www.cnplus.co.uk news s lucas ... External links http www.lucasuk.com Official company website http www.lucasfitout.com Lucas Fit Out ... more details
San Lucas the Spanish language name of St. Luke is sometimes used as a toponym Colombia Serrania of San Lucas mountains Costa Rica San Lucas Island Guatemala San Lucas Sacatep quez Sacatep quez department San Lucas Tolim n Solol department Honduras San Lucas, El Para so Mexico San Lucas, Chiapas San Lucas, Michoac n San Lucas Tecopilco , Tlaxcala San Lucas Quiavini , Oaxaca San Lucas Zoquiapam , Oaxaca Cabo San Lucas , Baja California Sur Nicaragua San Lucas, Madriz United States San Lucas, California San Lucas AVA , California wine region in Monterey County disambig de San Lucas es San Lucas gl San Lucas it San Lucas nl San Lucas pt San Lucas ru ... more details
volume 39 year 2001 . See also Somer Lucaspseudoprime Category Recurrence relations Category ...distinguish2 the sequence of Lucas number s, which is a particular Lucas sequence In mathematics , the Lucas ... of the Lucas sequences U sub n sub P , Q and V sub n sub P , Q . More generally, Lucas sequences U sub .... Famous examples of Lucas sequences include the Fibonacci number s, Mersenne number s, Pell number s, Lucas number s, Jacobsthal number s, and a superset of Fermat number s. Lucas sequences are named after the France French mathematician douard Lucas . Recurrence relations Given two integer parameters P and Q , the Lucas sequences of the first kind U sub n sub P , Q and of the second kind ... of Lucas sequences U sub n sub P , Q and V sub n sub P , Q are given in the table class wikitable ... The characteristic equation of the recurrence relation for Lucas sequences math U n P,Q math and math ... n sqrt D 2 math math b n frac V n U n sqrt D 2 math . It follows that the terms of Lucas sequences ... nS n 1 , math math V n P,Q V n 2S,S 2 2S n , math . Other relations The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci number s math F n U n 1, 1 math and Lucas ... computation of math U n P,Q math for large values of n . These facts are used in the Lucas Lehmer primality test . Carmichael s theorem states that all but finitely many of the terms in a Lucas sequence ... names The Lucas sequences for some values of P and Q have specific names U sub n sub 1,&minus 1 Fibonacci number s V sub n sub 1,&minus 1 Lucas number s U sub n sub 2,&minus 1 Pell number s V sub n sub 2,&minus 1 Companion Pell numbers or Pell Lucas numbers U sub n sub 1,&minus 2 Jacobsthal ... n sub x ,&minus 1 Fibonacci polynomials V sub n sub x ,&minus 1 Lucas polynomials Applications LUC is a public key cryptosystem based on Lucas sequences ref cite journal author P. J. Smith, M. J. J. Lennon ... . The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead ... more details
Lucas County is the name of two counties in the United States Lucas County, Iowa Lucas County, Ohio Templates geodis uscounty Interwiki de Lucas County es Condado de Lucas fr Comt de Lucas it Contea di Lucas nl Lucas County no Lucas County nds Lucas County pl Hrabstwo Lucas pt Condado de Lucas sv Lucas County uk zh ... more details
Lucas Silva may refer to Lucas da Silva Lucas , Brazilian footballer Lucas Ant nio Silva de Oliveira , Brazilian footballer Lucas Borges da Silva , Brazilian footballer Lucas Rodrigues Moura da Silva commonly known as Lucas born 1992 Brazilian international footballer hndis Silva, Lucas ... more details
Lucas Green may refer to Lucas Green, Lancashire , England Lucas Green, Surrey , England geodis Long comment to avoid being listed on short pages pl Lucas Green ... more details
Ben Lucas is the name of Ben Lucas rugby union born 1987 , Australian rugby union footballer Ben Lucas lobbyist , co owner of the lobbying firm LLM Communications hndis Lucas, Ben ... more details
Margaret Lucas may refer to Margaret Bright Lucas Margaret Cavendish, Duchess of Newcastle upon Tyne , nee Lucas hndis Lucas, Margaret ... more details
John Lucas may refer to John Lucas MP , British Member of Parliament for for Colchester UK Parliament constituency Colchester John Lucas philosopher born 1929 , British philosopher John Lucas II born 1953 , retired American professional basketball player John Lucas III born 1982 , his son, current professional basketball player John Meredyth Lucas 1919 2002 , screenwriter and director John Lucas VC 1826 1892 , Irish recipient of the Victoria Cross John P. Lucas 1890 1949 , American general in World War II John Lucas comics , American comic book artist John S. Lucas comics , American comic book artist John Baptiste Charles Lucas 1758 1842 , U.S. Congressman from Pennsylvania John Lucas cricketer 1922 2008 , West Indian and Canadian cricketer John Robert Lucas born 1961 , American Christian author John Seymour Lucas 1849 1923 , Victorian English historical and portrait painter Sir John Lucas royalist , brother of Charles Lucas Johnny Lucas baseball 1903 1970 , backup outfielder Johnny Lucas canoer born 1931 , Luxembourgian sprint canoer John Lucas Australian politician 1818 1902 John Lucas priest 1921 1992 , Archdeacon of Totnes See also Jon Lucas , American screenwriter St. John Lucas 1879 1934 , English poet hndis Lucas, John de John Lucas it John Lucas ... more details
David Lucas is the name of David Lucas composer born 1937 , American rock and roll producer and jingle writer David Lucas cricketer born 1978 , English cricketer who plays for Worcestershire David Lucas footballer born 1977 , English footballer Steven Blum born 1965 , American voice actor who used the alias David Lucas Dave Lucas born 1932 , Canadian ice hockey player hndis Lucas, David DEFAULTSORT Lucas, David it David Lucas ... more details
Lucas Cranach may refer to Lucas Cranach the Elder 1472&ndash 1553 Lucas Cranach the Younger 1515&ndash 1586 hndis Cranach, Lucas br Lucas Cranach cs Cranach de Lucas Cranach es Lucas Cranach fr Lucas Cranach hy it Lucas Cranach lv L kass Kr nahs nl Lucas Cranach no Lucas Cranach pl Lucas Cranach ru sr fi Lucas Cranach sv Lucas Cranach zh ... more details
For the Swiss bobsledder Michael Lukas Michael Lucas is the name of Michael Lucas director born 1972 , performer in and director of gay pornographic films Michael Lucas political activist born 1926 , artist, designer, and political activist Michael Lucas, 2nd Baron Lucas of Chilworth 1926 2001 , British peer and Conservative politician Mike Lucas , football coach hndis name Lucas, Michael th tr Michael Lucas ... more details
Henry Lucas is the name of Henry Lucas politician c. 1610 1663 , English MP, philanthropist and benefactor to Cambridge University Henry Lucas baseball 1858 1910 , American millionaire and baseball executive Henry Lee Lucas 1936 2001 , American convicted serial killer Henry Lucas, British car enthusiast, associated with James and Browne automobiles See also Lucas surname Lucas disambiguation hndis Lucas, Henry ja ... more details
Lucas test may refer to The Lucas primality test for primality of general numbers The Lucas Lehmer primality test for Mersenne prime Mersenne primes Lucas reagent , used to classify alcohols of low molecular weight disambig ... more details
Incoming links date February 2012 Robert Lucas may refer to Rob Lucas born 1953 , Liberal member of the South Australian Legislative Council Robert Lucas, 3rd Baron Lucas Robert Lucas, Jr. born 1937 , economist Robert Lucas governor 1781 1853 , governor of Ohio, 1832 36, first governor of the Iowa Territory 1838 41 Robert Lucas candidate , 2004 Republican candidate for U.S. Representative in Ohio Robert Slade Lucas 1867 1942 , cricketer Robert Lucas, musician in the band Canned Heat Robert Lucas MP , for Ipswich UK Parliament constituency See also Robert Lucas de Pearsall 1795 1856 , English composer hndis Lucas, Robert de Robert Lucas hr Robert Lucas it Robert Lucas ja tr Robert Lucas uk ... more details
Richard Lucas is the name of Richard Lucas clergyman c. 1640 1715 , Welsh clergyman Richard Lucas politician Richard Lucas rower 1886 1968 , British rower Dick N. Lucas 1920 1997 , American film animator Dick Lucas minister born 1925 , British evangelical speaker Richie Lucas born 1938 , American footballer Dick Lucas , British vocalist Richard Cockle Lucas 1800 1883 , English sculptor Dick Lucas, fictional character from Are You Being Served? hndis name Lucas, Richard fr Dick Lucas ... more details
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