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 Lucas pseudoprime Fibonacci pseudoprime Lucas pseudoprime Perrin pseudoprime Somer Lucas pseudoprime Strong Lucas pseudoprime 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 classes of Lucas pseudoprime 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 U 1 P,Q 1 , 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 Lucas pseudoprime ref ... Lucas pseudoprime is an odd composite number n with n,D 1, and n 2 sup r sup s with s odd, satisfying ... r . A strong Lucas pseudoprime is also a Lucas pseudoprime. An extra strong Lucas pseudoprime is a strong Lucas pseudoprime 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 Lucas pseudoprime is also a strong Lucas pseudoprime. By combining a Lucas pseudoprime test with a Fermat primality test , say, to base 2, one can obtain ... 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 ... FibonacciPseudoprime title Fibonacci Pseudoprime MathWorld urlname LucasPseudoprime title Lucas Pseudoprime MathWorld urlname StrongLucasPseudoprime title Strong Lucas Pseudoprime MathWorld urlname ExtraStrongLucasPseudoprime title Extra Strong Lucas Pseudoprime Category Fibonacci numbers Category ... 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
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 Lucas Pseudoprime 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 Lucas Pseudoprime Category Pseudoprimes ... more details
prime which is composite is called an Euler&ndash Jacobi pseudoprime to base a . This test ... prime to base a is called a strong pseudoprime to base a . Every strong probable prime to base ... pseudoprime Carmichael number Miller Rabin primality test Provable prime External links http primes.utm.edu ... more details
The Fibonacci numbers are the best known concept named after Leonardo of Pisa , known as Fibonacci . Among others are the following. Concepts in mathematics and computing Brahmagupta Fibonacci identity Fibonacci coding Fibonacci cube Fibonacci heap Fibonacci polynomials Fibonacci prime Fibonacci pseudoprime Fibonacci quasicrystal Fibonacci retracement Fibonacci search technique Fibonacci triangle Fibonacci Sylvester expansion Fibonacci word Lagged Fibonacci generator Negafibonacci NegaFibonacci coding Pisano period Reciprocal Fibonacci constant Young Fibonacci lattice A professional association and a scholarly journal that it publishes The Fibonacci Association Fibonacci Quarterly An asteroid 6765 Fibonacci An art rock band The Fibonaccis Category Lists of things named after mathematicians Fibonacci ... more details
In number theory , the Baillie PSW primality test is a Randomized algorithm probabilistic primality test ing heuristic algorithm it determines if a number is composite number composite or a probable prime . The authors of the test offered 30 for the discovery of a composite number that passed this test. As of 1994 , the value was raised to 620 ref Richard K. Guy Guy, R. 1994 . Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes. A12 in Unsolved Problems in Number Theory . 2nd ed., p. 28, New York Springer Verlag. ISBN 0 387 20860 7. ref and no pseudoprime was found up to 10 sup 17 sup , ref MathWorld urlname Baillie PSWPrimalityTest title Baillie PSW Primality Test ref consequently this can be considered a sound primality test on numbers below that upper bound. A primality testing software PRIMO ref http www.ellipsa.net primo index.html PRIMO ref uses this algorithm to check for probable prime s, and no certification of this test has yet failed. The author, Marcel Martin, estimates by those results that the test is accurate for numbers below 10000 digits. There is a heuristic argument harv Pomerance 1984 suggesting that there may be infinitely many counterexamples. ref citation authorlink Carl Pomerance last Pomerance first C. year 1984 url http www.pseudoprime.com dopo.pdf title Are There Counterexamples to the Baillie PSW Primality Test? ref The test Optionally, perform trial division to check if the number isn t a multiple of a small prime number . Perform a base 2 strong pseudoprime strong pseudoprimality test . If it fails, n is composite. Find the first a in the sequence 5, &minus 7, 9, &minus 11, ... for which the Jacobi symbol math left frac a n right 1 math . Perform a Lucas pseudoprime Lucas pseudoprimality test with discriminant a on n . If this test does not fail, n is likely a prime. References reflist Further reading citation first Thomas R. last Nicely url http www.trnicely.net misc bpsw.html title The Baillie PSW primality test. citation authorlink ... more details
Unreferenced date December 2009 width 250 border 1 style float right background feffff border collapse collapse margin 0.5em colspan 2 List of numbers br 10000 number 10000 64079 100000 number 100000 Cardinal number Cardinal 64079 br Sixty four thousand and seventy nine. Ordinal number Ordinal 64079th br Sixty four thousand and seventy ninth. Factorization 139 461 Divisor s 1, 139, 461, 64079 Roman numeral Overline L Overline X MMMMLXXIX Binary numeral system Binary 1111101001001111 Octal 175117 Hexadecimal FA4F 64079 is the twenty third Lucas number and is thus often written as L sub 23 sub . It is significant for being the first Lucas number L sub n sub where n is prime number prime that is itself composite number not prime . In a closely related property, 64079 is the smallest extra strong Lucas pseudoprime to base 2. Its smaller prime factor, 139 number 139 , can be written in the form 6 n 1, where n is the index of the Lucas number. The larger factor, 461, can similarly be written in the form 20 n 1. Other uses 64079 is the zip code of Platte City, Missouri Platte City and Tracy, Missouri Tracy , Missouri . DEFAULTSORT 64079 Number Category Integers 69e04 64079 ... more details
Probabilistic algorithm Fermat primality test Pseudoprime Carmichael number Euler pseudoprime Euler Jacobi pseudoprime Fibonacci pseudoprime Probable prime Miller Rabin primality test Lucas Lehmer ... more details
Pierre Fr d ric Sarrus 10 March 1798, Saint Affrique 20 November 1861 was a France French mathematician . Sarrus was professor at the University of Strasbourg , France 1826 1856 and member of the French Academy of Sciences in Paris 1842 . He is author of several treatises, including one on the solution of numeric equations with multiple unknowns 1842 one on multiple integrals and their integrability conditions one on the determination of the orbits of the comets . He also discovered a mnemonic rule for solving the determinant of a 3 by 3 matrix mathematics matrix , named Rule of Sarrus Sarrus scheme , which provides an easy to remember method of working out the determinant of a 3 by 3 matrix as illustrated in cross product . Sarrus also demonstrated the fundamental lemma mathematics lemma of the calculus of variations . Sarrus number s are pseudoprime s to base 2. Sarrus also developed the Sarrus linkage , the first linkage capable of transforming rotary motion into perfect straight line motion, and vice versa. Persondata Metadata see Wikipedia Persondata . NAME Sarrus, Pierre Frederic ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 10 March 1798 PLACE OF BIRTH DATE OF DEATH 20 November 1861 PLACE OF DEATH DEFAULTSORT Sarray, Pierre Frederic Category French mathematicians Category 19th century mathematicians Category 1798 births Category 1861 deaths France mathematician stub de Pierre Fr d ric Sarrus es Pierre Fr d ric Sarrus fr Pierre Fr d ric Sarrus it Pierre Fr d ric Sarrus ht Pierre Frederic Sarrus pl Pierre Fr d ric Sarrus pt Pierre Fr d ric Sarrus ru , sr ... more details
italic title Infobox Journal title Rejecta Mathematica cover editor Michael Wakin, Christopher Rozell, Mark Davenport, Jason Laska discipline mathematics language English language English abbreviation publisher country frequency history openaccess impact impact year website http www.rejecta.org link1 link1 name link2 link2 name RSS atom JSTOR OCLC 402335845 LCCN CODEN ISSN 1948 8351 eISSN Rejecta Mathematica is an online Academic journal journal for publishing Academic publishing Scholarly paper paper s that have been rejected by other mathematics journals such as Annals of Mathematics . Each paper is accompanied by an open letter describing why the paper was rejected, how the topic has been developed since and why it is worthy of publication. The first issue was published in July 2009 containing topics such as image enhancement and condition number s. ref name Hm The quality of the contributions is mixed but one paper has already attracted more citations than another paper which the same author had published in the Journal of the American Mathematical Society . ref name AT citation url http arstechnica.com science news 2009 07 send us your rejects rejecta mathematica goes live.ars title Send us your rejects author John Timmer date July 20, 2009 publisher Ars Technica ref The editors are Michael Wakin, Christopher Rozell, Mark Davenport and Jason Laska. ref name Hm citation url http www.economist.com sciencetechnology displaystory.cfm?story id 14119761 title Huddled maths publisher The Economist date August 1, 2009 volume 392 issue 8642 ref After almost two years since the inaugural issue, the second issue was published in June 2011 and contains topics such as subspace classification and distributions of pseudoprime s. See also Deletionpedia References reflist External links http www.rejecta.org Rejecta Mathematica Category Mathematics journals Category Publications established in 2009 academic journal stub ... more details
, math 2 341 equiv 2 pmod 341 , math , but 341 11 31 is a pseudoprime . See ... , then p need not be prime. If it is not, then p is called a pseudoprime to base a . F. Sarrus in 1820 found 341 11 31 as one of the first pseudoprimes, to base 2. A number p that is a pseudoprime ... more details
strong prime. Note that the criteria for determining if a pseudoprime is a strong pseudoprime is by congruences to powers of a base, not by inequality to the arithmetic mean of neighboring ... more details
In mathematics , an integer sequence is a sequence i.e., an ordered list of integer s. An integer sequence may be specified explicitly by giving a formula for its n th term, or implicitly by giving a relationship 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 to the formula n sup 2 sup &minus 1 for the n th term an explicit definition. Alternatively, an integer sequence 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 which have received their own name include div class style moz column count 3 column count 3 Abundant number s Baum Sweet sequence Bell number s Binomial coefficient s Carmichael number s Catalan number s Composite number s Deficient number s Euler number s Even and odd numbers Factorial numbers Fibonacci 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 integer sequence is a Recursion theory computable sequence , if there exists an algorithm which given n , calculates a sub n sub , for all n > 0. An integer sequence is a definable set definable sequence , if there exists some statemen ... more details
105 one hundred and five is the natural number following 104 number 104 and preceding 106 number 106 . Number number 105 range 100s cardinal one hundred and five ordinal th ordinal text one hundred and fifth numeral 105 factorization math 3 cdot 5 cdot 7 math prime divisor 1, 3, 5, 7, 15, 21, 35, 105 roman CV unicode greek prefix latin prefix bin 1101001 oct duo hex 69 misc In mathematics 105 is a triangular number , a polygonal number 12 gonal number and a Zeisel number . It is a sphenic number , and is the product of three consecutive prime number s. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal number s. 105 comes in the middle of the prime quadruplet 101, 103, 107, 109 . The only other such odd numbers less than a thousand are 15, 195 and 825. 105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth Aaron pair under the first definition. 105 is also a number n for which math n 2 k math is prime, for math 0 k log 2 n math . This even works up to math k 8 math , ignoring the negative sign. 105 is the smallest integer such that the factorization of math x n 1 math over Q includes coefficients other than math pm 1 math . In science The atomic number of hahnium , also known as dubnium . In other fields 105 is also The number of sura t al Fil in the Qur an . Citation needed date May 2010 A simple card game where players lose if they have over 105 points. See also List of highways numbered 105 References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London Penguin Group. 1987 134 DEFAULTSORT 105 Number Category Integers ar 105 az 105 d d ca Cent cinc cv 105 eu Ehun eta bost fa fr 105 nombre ko 105 id 105 angka it 105 numero ht 105 nonm hu 105 sz m mk 105 ms 105 nombor nl 105 getal ja 105 no 105 tall pt Cento e cinco ru 105 nso 105 nomoro sl 105 tevilo s ... more details
In number theory , the Chinese hypothesis is a disproven conjecture stating that an integer n is Prime number prime if and only if it satisfies the condition that 2 sup n sup &minus 2 is divisible by n . In other words, that integer n is prime if and only if math 2 n equiv 2 pmod n , math . It is true that if n is prime, then math 2 n equiv 2 pmod n , math this is a special case of Fermat s little theorem . However, the converse if math ,2 n equiv 2 pmod n math then n is prime is false, and therefore the hypothesis as a whole is false. The smallest counter example is n 341 11 31. Composite number s n for which 2 sup n sup &minus 2 is divisible by n are called Poulet number s. They are a special class of Fermat pseudoprime s. History The Chinese hypothesis is commonly attributed to Chinese mathematics Chinese scholars more than 2500 years ago. However, this oft quoted attribution is a myth originating with James Jeans 1898 , who wrote that a paper found among those of the late Sir Thomas Wade and dating from the time of Confucius contained the theorem . This assertion was refuted by Needham, who attributes the misunderstanding to an incorrect translation of a passage in a well known book The Nine Chapters of Mathematical Art . Qi 1991 attributed the hypothesis to Chinese mathematician Li Shanlan 1811 1882 , communicated the statement to his collaborator in the translation of Western texts, and the collaborator then published it. Li subsequently learned that the statement was wrong, and hence did not publish it himself, but Hua Heng Fang published the statement as if it were correct in 1882. References Citation last Dickson first L. E. title History of the Theory of Numbers, Vol. 1 Divisibility and Primality location New York publisher Dover year 2005 isbn 0486442322 . Citation last Erdos first P. title On the Converse of Fermat s Theorem journal American Mathematical Monthly volume 56 issue 9 pages 623 624 year 1949 doi 10.2307 2304732 . Citation last Honsberger firs ... more details
is that if math a n 1 not equiv 1 pmod n math i.e., n is not a pseudoprime to base a then no such information ... might try to send you a pseudoprime in a place where a prime number is required. In such cases ... Miller Strong Pseudoprime Test http gandraxa.com miller rabin primality test.xml Interactive Online ... more details
The Fermat primality test is a randomized algorithm probabilistic test to determine if a number is probable prime . Concept Fermat s little theorem states that if p is prime and math 1 le a p math , then math a p 1 equiv 1 pmod p . math If we want to test if p is prime, then we can pick random a s in the interval and see if the equality holds. If the equality does not hold for a value of a , then p is composite. If the equality does hold for many values of a , then we can say that p is probable prime . It might be in our tests that we do not pick any value for a such that the equality fails. Any a such that math a n 1 equiv 1 pmod n math when n is composite is known as a Fermat liar . Vice versa, in this case n is called Fermat pseudoprime to base a . If we do pick an a such that math a n 1 not equiv 1 pmod n math then a is known as a Fermat witness for the compositeness of n . Example Suppose we wish to determine if n 221 is prime. Randomly pick 1 a 221, say a 38. Check the above equality math a n 1 38 220 equiv 1 pmod 221 . math Either 221 is prime, or 38 is a Fermat liar, so we take another a , say 26 math a n 1 26 220 equiv 169 not equiv 1 pmod 221 . math So 221 is composite and 38 was indeed a Fermat liar. Algorithm and running time The algorithm can be written as follows Inputs n a value to test for primality k a parameter that determines the number of times to test for primality Output composite if n is composite, otherwise probably prime repeat k times pick a randomly in the range 1, n &minus 1 if math a n 1 not equiv1 pmod n math , then return composite return probably prime Using fast algorithms for modular exponentiation , the running time of this algorithm is Big O notation O k log sup 2 sup n log log n log log log n , where k is the number of times we test a random a , and n is the value we want to test for primality. Flaw There are infinitely many values of math n math known as Carmichael number s for which u all u values of mat ... more details
91 ninety one is the natural number following 90 number 90 and preceding 92 number 92 . Number number 91 range 90s cardinal ninety one ordinal st ordinal text ninety first numeral 91 factorization math 7 cdot 13 math prime divisor 1, 7, 13, 91 roman XCI unicode greek prefix latin prefix bin oct duo hex 5B misc wiktionary ninety one In mathematics Ninety one is the twenty seventh distinct semiprime and the second of the form 7.q . The aliquot sum of 91 is 21 within the aliquot sequence 91,21,11,1,0 91 being the fourth composite number in the 11 aliquot tree. Ninety one is a triangular number and a hexagonal number , one of the few such numbers to also be a centered hexagonal number , and it is also a centered nonagonal number and a centered cube number . It is a square pyramidal number , being the sum of the squares of the first six integer s. It is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed alternatively the sum of two cubes and the difference of two cubes 91 6 sup 3 sup 5 sup 3 sup 4 sup 3 sup 3 sup 3 sup . See 1729 number 1729 for more details . It is also the smallest positive integer expressible as a sum of six distinct squares 91 1 sup 2 sup 2 sup 2 sup 3 sup 2 sup 4 sup 2 sup 5 sup 2 sup 6 sup 2 sup . The only other ways to write 91 as a sum of distinct squares are 91 1 sup 2 sup 4 sup 2 sup 5 sup 2 sup 7 sup 2 sup and 91 1 sup 2 sup 3 sup 2 sup 9 sup 2 sup . It is also the smallest pseudoprime satisfying the congruence 3 sup n sup 3 mod n . ref Friedman, Erich. http www.stetson.edu efriedma numbers.html What s Special About This Number? ref 91 is a repdigit in base 9 111 . In science The atomic number of protactinium , an actinide. McCarthy 91 function , a recursive function in discrete mathematics In astronomy , Messier object Messier 91 M91 , a visual magnitude magnitude 11.5 spiral galaxy in the constellation Coma Berenices The New General Catalogue http www.ngcic.org object NGC 91 , ... more details
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