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Topic: Coprime


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  Coprime - Wikipedia, the free encyclopedia
In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1.
For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3.
The two integers a and b are coprime if and only if the point with coordinates (a, b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b).
en.wikipedia.org /wiki/Coprime   (550 words)

  
 Coprime -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21)
The number (The smallest whole number or a numeral representing this number) 1 is coprime to every integer; (A mathematical element that when added to another number yields the same number) 0 is coprime only to 1 and −1.
In fancy language: a and b are coprime if and only if b yields a (An organization regarded as part of a larger social group) unit in the (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring Z
As a consequence, if a and b are coprime and br ≡ bs ((A British teenager or young adult in the 1960s; noted for their clothes consciousness and opposition to the rockers) mod a), then r ≡ s (mod a) [because we may "divide by b" when working modulo a].
www.absoluteastronomy.com /encyclopedia/c/co/coprime.htm   (605 words)

  
 Coprime: Definition and Links by Encyclopedian.com - All about Coprime   (Site not responding. Last check: 2007-10-21)
In mathematics, the integers a and b are said to be coprime or relatively prime iff they have no common factor other than 1 and -1, or equivalently, if their greatest common divisor is 1.
If a and b are coprime and bx ≡ by (mod a), then x ≡ y (mod a).
The two integers a and b are coprime if and only if the point with coordinates (a,b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a,b).
www.encyclopedian.com /co/Coprime.html   (348 words)

  
 Coprime   (Site not responding. Last check: 2007-10-21)
In mathematics, the integerintegers ''a'' and ''b'' are said to be coprime''' or '''relatively prime iffif and only if they have no common divisorfactor other than 1 and −1, or equivalently, if their greatest common divisor/ is 1.
As a consequence, if ''a'' and ''b'' are coprime and ''br'' ≡ ''bs'' (modular arithmeticmod/ ''a''), then ''r'' ≡ ''s'' (mod ''a'') [because we may "divide by ''b''" when working modulo ''a''].
The two integers ''a'' and ''b'' are coprime if and only if the point with coordinates (''a'', ''b'') in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (''a'', ''b'').
www.infothis.com /find/Coprime   (445 words)

  
 PlanetMath: coprime
are coprime if their greatest common divisor is
This is version 4 of coprime, born on 2001-10-15, modified 2001-11-16.
Object id is 225, canonical name is Coprime.
planetmath.org /encyclopedia/Coprime.html   (35 words)

  
 Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
Note that, when the moduli are not pairwise coprime, some potential sets of "remainders" are ruled out: For example, no integer can leave a remainder of 2 when divided by 6 and a remainder of 3 when divided by 4...
Modulo 10, for example, the reciprocal of 7 is 3, whereas 1 and 9 are their own reciprocals (the residues 0,2,4,5,6,8 are not coprime to 10 and have therefore no reciprocal modulo 10).
There are 72 residues coprime to 91 (72 is the Euler totient of 91).
home.att.net /~numericana/answer/modular.htm   (3088 words)

  
 Euler's Totient Function   (Site not responding. Last check: 2007-10-21)
(Coprime is defined this way: two numbers are coprime when they share no factor other than 1.) Note that φ(1)=1, because 1 is coprime to itself.
By "proper coprimes" I mean positive numbers coprime to n that don't exceed n.
Suppose a is a number coprime to n, and b and c are two different numbers, also coprime to n, in the range [1,n-1] and ab=ac, mod n.
mcraeclan.com /MathHelp/BasicNumberTotient.htm   (1172 words)

  
 Coprime Definition / Coprime Research   (Site not responding. Last check: 2007-10-21)
One reason is that mathematical knowledge is revised and updated in a different way; though arguably founded on experiment in some manner, it is not comparable to the natural sciences in this respect.
[click for more], the integers a and b are said to be coprime or relatively prime if and only if they have no common factorIn mathematics, a divisor of an integer n, additionally named a factor of n, is an integer which evenly divides n without leaving a remainder.
Two numbers are named coprime or relatively prime if their greatest common d...
www.elresearch.com /Coprime   (285 words)

  
 Coprime Partitions   (Site not responding. Last check: 2007-10-21)
Let F(N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts.
Notice that if N is the smallest integer such that s(N)=k then N-1+s(N) is the sum of the first k primes.
For example, the first occurrence of s(N)=5 in the above table is for N=24, so we have (24-1+5) = 28 = 2 + 3 + 5 + 7 + 11 This just expresses the fact that the smallest saturated partition into k coprime parts is just the smallest partition into k distinct primes.
www.mathpages.com /home/kmath194.htm   (286 words)

  
 [No title]
Since a,b are coprime whereas the sum of their cubes is even, they must both be odd.
KB> Since a,b are coprime whereas the sum of their cubes is even, they KB> must both be odd.
If a,b are coprime then a=u+v and b=u-v where u and v are non-zero.
www.math.niu.edu /~rusin/known-math/95/fermat.236   (992 words)

  
 Pythagorean Triples with Coefficients
Assume all coefficients are squarefree, whence the coprime lemma applies.
The coprime lemma shows x and y are odd, and a mod 4 argument shows z is odd.
By the coprime lemma, 3 cannot divide x or y, and x and z are odd.
www.mathreference.com /num-zext,ptwc.html   (1826 words)

  
 On x^2 + y^3 = z^6   (Site not responding. Last check: 2007-10-21)
Clearly if any two of x,y,z have a common factor then they also share a common factor with the third, so we can divide until the three numbers x,y,z are coprime.
Since a,b are coprime with unequal magnitudes, and the sum of their cubes is even, they must both be odd.
This implies the existence of coprime integers u,v such that a=u+v and b=u-v.
www.mathpages.com /home/kmath213.htm   (257 words)

  
 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-21)
What this means for this proof is that if we let y be the largest multiple of 6 less than or equal to x, then the number of numbers coprime to 6 which are less than or equal to y is 2*(y/6) = y/3.
Therefore, there are at most 2 numbers coprime to 6 between x and y.
On the other hand, there may be no numbers coprime to 6 between x and y.
mathforum.org /library/drmath/view/51977.html   (415 words)

  
 ECS EPrints Service - Nonlinear coprime factorizations and parameterization of a class of stabilizing controllers   (Site not responding. Last check: 2007-10-21)
Nonlinear coprime factorizations and parameterization of a class of stabilizing controllers
Baramov, L. and Kimura, H. Nonlinear coprime factorizations and parameterization of a class of stabilizing controllers.
New definitions for right,left and doubly coprime factorizations for nonlinear, input-affine state-space systems are introduced.
eprints.ecs.soton.ac.uk /2462   (172 words)

  
 Minimal Degree Coprime Factorization of Rational Matrices
Given a rational matrix G with complex coefficients and a domain $\Gamma$ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over $\Gamma$, with denominators of McMillan degree as small as possible.
The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G.
Further, we consider the case in which the denominator has a certain symmetry, namely it is J all-pass with respect either to the imaginary axis or to the unit circle.
epubs.siam.org /sam-bin/dbq/article/33997   (293 words)

  
 Maybe this Explains the Economic Cycle... best Coprime   (Site not responding. Last check: 2007-10-21)
integers a and b are said to be coprime or relatively prime iff they have no common factor...
Coprime was founded by alumni of the Massachusetts Institute of Technology from...
For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible...
ascot.pl /th/Fourier3/Coprime.htm   (392 words)

  
 Algebraic Construction of Normalized Coprime Factors for Delay Systems (ResearchIndex)   (Site not responding. Last check: 2007-10-21)
A parametrization is given of all the possible factorizations that can be obtained by solving algebraic equations over the eld generated by s and e s.
This enables us to provide a means of determining when such factors can be calculated explicitly, and to show that in general they cannot.
0.4: On Cycles in the Coprime Graph of Integers - Erdös, Sarkozy (1996)
citeseer.ist.psu.edu /295907.html   (405 words)

  
 Math Refresher: CoPrime Numbers: xn + yn = zn
(b) it is possible to cancel out common components and derive three numbers that are coprime.
Two numbers are coprime if they do not share any common divisors.
Two numbers x,y are said to not be coprime if and only if there exists a value d > 1 such that d divides x, d divides y
mathrefresher.blogspot.com /2005/05/coprime-numbers-xn-yn-zn.html   (354 words)

  
 Paj's Home: Cryptography: RSA: Mathematics
A pair of numbers are coprime if the largest number that exactly divides both of them is 1.
8 and 9 are coprime, but 8 and 10 are not, because they're both divisible by 2.
If you have a pair of distinct prime numbers, they will always be coprime to each other.
pajhome.org.uk /crypt/rsa/maths.html   (553 words)

  
 The CTK Exchange Forums
"The fact is that for m and n coprime of different parities, (*) yields coprime numbers a, b, and c.
Conversely, all coprime triples can indeed be obtained in this manner.
2nm is 72 so nm is 36, the only coprime divisiors are 1 and 36.
www.cut-the-knot.com /htdocs/dcforum/DCForumID5/427.shtml   (137 words)

  
 Coprime Factorizations and Well-Posed Linear Systems
Coprime Factorizations and Well-Posed Linear Systems: SIAM Journal on Control and Optimization Vol.
We first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in $H^\infty$.
The converse is also true: every function with a doubly coprime factorization in $H^\infty$ is the transfer function of a jointly stabilizable and detectable well-posed linear system.
epubs.siam.org /sam-bin/dbq/article/28541   (180 words)

  
 PlanetMath: Fermat Numbers are coprime
Therefore, the greatest common divisor of any two Fermat numbers must be
Q.E.D. "Fermat Numbers are coprime" is owned by drini.
This is version 1 of Fermat Numbers are coprime, born on 2004-11-25.
planetmath.org /encyclopedia/FermatNumbersAreCoprime.html   (59 words)

  
 Coprime Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-10-21)
Looking For coprime - Find coprime and more at Lycos Search.
Find coprime - Your relevant result is a click away!
Look for coprime - Find coprime at one of the best sites the Internet has to offer!
www.localcolorartists.com /encyclopedia/Coprime   (663 words)

  
 The Coprime Lemma
This continues until the variables are pairwise coprime.
If the coefficients are pairwise coprime, which is not the case in the example shown above, each coefficient is coprime to the other two variables.
In the smallest solution, variables are coprime to each other, and each variable is coprime to the other two coefficients.
www.mathreference.com /num-zext,copl.html   (609 words)

  
 PlanetMath: mutually coprime
are called mutually coprime if there is no
would be mutually coprime though they are not pairwise coprime, because that would mean that each pair of these numbers is
This is version 2 of mutually coprime, born on 2004-04-14, modified 2004-04-14.
planetmath.org /encyclopedia/MutuallyCoprime.html   (49 words)

  
 [No title]   (Site not responding. Last check: 2007-10-21)
In this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view.
For linear systems there are several ways to approach this problem, and we generalize the ideas of balancing the normalized coprime factorization and of LQG balancing to the nonlinear case.
LQG balancing can be seen as measuring the influence of a state component on a certain future and past energy function.
dutera.et.tudelft.nl /~scherpen/abstracts/ijc94.htm   (203 words)

  
 Probability of two numbers being Coprime   (Site not responding. Last check: 2007-10-21)
The probability that two independently chosen large numbers are both divisible by p is then 1/p^2.
If the two numbers are coprime, then for each prime p, p does not divide both numbers.
, so that the reciprocal of this is the probability of two large numbers being coprime.
www.ballandclaw.com /upi/coprime.html   (317 words)

  
 Coprime Factor Reduction Of (ResearchIndex)
Abstract: We consider the efficient solution of the coprime factorization based controller approximation problems by using frequency-weighted balancing related model reduction approaches.
It is shown that for a class of frequency-weighted performance preserving coprime factor reduction as well as for a relative error coprime factor reduction method, the computation of the frequency-weighted controllability and observability grammians can be done by solving Lyapunov equations of the order of the...
9 Coprime factorization controller reduction with Bezout ident..
citeseer.ist.psu.edu /656426.html   (388 words)

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