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 0 , math The , is to keep the formula rendered as PNG instead of HTML. Please don t remove it. math ... ne 0 math , then p is a factor of U sub p sub . Lucas pseudoprimes A Lucaspseudoprime is a composite ... of 377. A strong Lucaspseudoprime is an odd composite number n with n,D 1, and n 2 sup r sup s with s odd ... for some j < 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 ... V sub 2 sup j sup s sub for some j < r . An extra strong Lucaspseudoprime is also a strong Lucaspseudoprime. 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 that in this case ... under 2,217,967,487 and their factors. MathWorld urlname 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 Pseudoprimes fr Nombre pseudopremier de ... Lucas pseudoprimes are 323 and 377 math left frac 5 323 right math and math left frac 5 377 right ... Fibonacci pseudoprime is 443372888629441, which has factors 17, 31, 41, 43, 89, 97, 167 and 331 ... 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 http books.google.com books?id 72eg8bFw40kC&pg PA131 2.X.D Somer Lucas Pseudoprimes 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 which is not actually prime. Pseudoprimes can be classified according to which property they satisfy. 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 col end References reflist Category Pseudoprimes ca Nombre pseudoprimer cs Pseudoprvo slo de Pseudoprimzahl es N mero pseudoprimo eo Pse doprimo fr Nombre pseudopremier it Pseudoprimo nl Kleine stelling van Fermat Pseudo priemgetallen pl Liczby pseudopierwsze pt Pseudoprimo ru sl Psevdopra tevilo fi N enn isalkuluku vi S gi nguy n t zh ... 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 Frobenious 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 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 iff 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. However, Richard K. Guy Guy uses a definition with only the first condition 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 . Because not all primes pass that condition for all bases, this definition of strong pseudoprimes is less generally applicable. 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 ... to all bases. Thus given a random base, the probability that a number is a strong pseudoprime to that base ... 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 pseudoprimes to base a if it successfully passes Fermat primality test for the base a . 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 to x is called a Carmichael number . Variations Some sources use variations of the definition, for example to only allow odd numbers to be pseudoprimes. ref MathWorld 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 given by Richard Crandall Crandall and Carl Pomerance Pomerance ref cite book author Richard Crandall, Carl Pomerance title Prime Numbers A Computational Perspective ... pseudoprime to a base a , if math a q 1 equiv 1 pmod q math and math 2 le a le q 2. math This stronger ... The smallest pseudoprime for each base a 200 is given in the following table the colors mark ... 67 Euler Jacobi pseudoprimes main Euler 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 ... small chance that the number found is not a prime number but a pseudoprime, it is possible to use ... more details
wiktionarypar Lucas TOCright Lucas may refer to several things. People bearing the surname Lucas See Lucas surname People bearing the given name Lucas Real Lucas Biglia , Argentine footballer currently playing for RSC Anderlecht Lucas Bolsius , Dutch historian and politician 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 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 Vita , Brazilian water polo player Jefferson Lucas Azevedo dos Santos , Brazilian footballer 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 in the soap opera Home and Away Lucas Johnson , character in the BBC soap opera EastEnders Lucas Kane , main character in the Fahrenheit video game Lucas Roberts , character on the NBC soap opera Days of our Lives Lucas Ross, character in Evernight , by Claudia Gray Lucas Scott , teen character on One Tree Hill Lucas Valieri , a character in Degrassi The Next Generation Lucas Wolenczak , teenage character on SeaQuest DSV Industry Lucas Industries , manufacturer of components for the motor and aerospace industry. Places In the United States Lucas, Iowa Lucas County, Iowa Lucas, Kansas Lucas, Ohio Lucas, Texas Lucas, Wisconsin ... more details
Lucas County is the name of two counties in the United States Lucas County, Iowa Lucas County, Ohio Templates geodis 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 ro Comitatul Lucas sv Lucas County zh ... more details
Scanned 39 5 yabuta.pdf volume 39 year 2001 . See also Somer Lucaspseudoprime Category Recurrence ...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 . Famous examples of Lucas sequences include the Fibonacci number s, Fermat number s, Mersenne number s, Pell number s, Lucas numbers and Jacobsthal 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 ... 2 Q 2 2 Q 3 , math Algebraic relations The characteristic equation of the recurrence relation for Lucas ... that the terms of Lucas sequences can be expressed in terms of a and b as follows math U n frac a n ... 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 number s math L n V n 1, 1 math . For example class wikitable ... that allows fast computation of math U n math for large values of n . These facts are used in the Lucas Lehmer primality test . Carmichael s theorem states that, in a Lucas sequence, all but finitely ... harv Yubuta 2001 . Specific 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 ... Yubuta 2001 . Applications LUC is a public key cryptosystem based on Lucas sequences ref cite journal ... LUCDIF , and RSA LUCRSA . The encryption of the message in LUC is computed as a term of certain Lucas ... Remarks on Lucas Based Cryptosystems journal Lecture Notes in Computer Science volume 963 year 1995 ... 0883853337 page 35 MathWorld urlname LucasSequence title Lucas Sequence Wei Dai. http weidai.com lucas.html Lucas Sequences in Cryptography . citation last Yabuta first M. journal Fibonacci Quarterly ... 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 it San Lucas nl San Lucas pt San Lucas ru ... 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 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
Lucas as a surname may refer to Albert Lucas , American juggler Anthony Francis Lucas 1855 1921 , Croatian born oil field engineer Antoinette Lucas born 1968 , American field hockey player Al Lucas 1978 2005 , American football player Al Lucas musician 1916 1983 , Canadian jazz double bassist A. P. Lucas 1857 1923 , English cricketer Baron Lucas, aristocratic family name see below Arthur Lucas , one of the last two people to be executed in Canada Arthur Lucas academic , 18th Principal of King s College London Arthur Henry Shakespeare Lucas 1853 1936 , English born Australian schoolmaster and scientist Buddy Lucas , New Zealand swimmer Caroline Lucas born 1960 , British politician Charles Lucas 1613 1648 , English soldier Charles Lucas musician 1808 1869 , British cellist Charles Lucas politician 1713 1771 , Irish politician and physician Charles Davis Lucas 1834 1914 , Anglo Irish soldier, Victoria Cross recipient Charlotte Lucas born 1976 , British actress Clarence Lucas 1866 1947 , Canadian composer Colin Lucas born 1940 , British historian Craig Lucas born 1951 , American playwright David Lucas, pseudonym of Steven Blum born 1965 , American voice actor Dick Lucas minister born 1925 , British evangelical speaker Dick Lucas , British vocalist douard Lucas 1842 1891 , French mathematician Edward Lucas disambiguation Edward Lucas , one of several people including Edward Lucas director , American film director Edward Lucas journalist , British journalist E. V. Lucas Edward Verrall Lucas 1868 1938 , British essayist Eliza Lucas c.1722 1793 , Antiguan born plantation manager Francis Lucas disambiguation Francis Lucas , various including Francis Lucas Royal Navy officer c. 1741 1770 , naval officer and merchant trader born in Clontibret, Ireland Francis Lucas Irish politician ... Borough 1713 1746 and Monaghan County Francis Lucas English politician 1850 1918 , British company director and Conservative Member of Parliament for Lowestoft 1900 1906 douard Lucas Fran ois douard ... more details
Lucas test may refer to The Lucas primality test for primality of general numbers The Lucas Lehmer primality test for Mersenne prime Lucas reagent , used to classify alcohols of low molecular weight disambig ... more details
Frank Lucas is the name of Frank E. Lucas 1876 1948 , American politician F. L. Lucas 1894 1967 , English literary critic Frank Lucas drug lord born 1930 , American drug dealer Frank Lucas Oklahoma born 1960 , American politician Frank Lucas boxer , ABA Middleweight Champion See also Francis Lucas disambiguation hndis Lucas, Frank de Frank 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
John Lucas is the name of John Lucas philosopher born 1929 , 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 1827 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 born 1922 , 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 1932 , backup outfielder Johnny Lucas canoer born 1931 , Luxembourgian sprint canoer John Lucas Australian politician 1818 1902 , nineteenth century Australian politician See also Jon Lucas , American screenwriter St. John Lucas 1879 1934 , English poet hndis Lucas, John de John Lucas it John Lucas ... more details
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 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 ... more details
Edward Lucas is the name of Edward Lucas congressman 1780 1858 , United States Congressman from Virginia Edward Lucas Irish politician 1787 1871 , Member of Parliament for Monaghan UK Parliament constituency Monaghan 1834 1841 douard Lucas 1842 1891 , French mathematician E. V. Lucas 1868 1938 , British author Edward Lucas director , American film director Edward Lucas journalist born 1962 , British journalist Ed Lucas born 1939 , sportswriter Ed Lucas baseball born 1982 Eddie Lucas , basketball player hndis name Lucas, Edward ... 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 ... more details
David Lucas is the name of David Lucas cricketer born 1978 , English cricketer who plays for Northamptonshire 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
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
Encyclopedia
top
