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| {{wiktionary|coprime}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Some time ago he chose to reside in North Carolina and he doesn't strategy on changing it. To climb is something I truly appreciate doing. Invoicing is my occupation.<br><br>Feel free to visit my homepage: online psychics ([http://youulike.com/profile/190212 http://youulike.com/]) |
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| In [[number theory]], two [[integer]]s ''a'' and ''b'' are said to be '''relatively prime''', '''mutually prime''', or '''coprime''' (also spelled '''co-prime''')<ref>Eaton, James S. Treatise on Arithmetic. 1872. May be downloaded from: http://archive.org/details/atreatiseonarit05eatogoog</ref> if the only positive integer that evenly divides both of them is 1 - they have no common positive factors other than 1. This is equivalent to their [[greatest common divisor]] being 1.<ref>{{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=[[E. M. Wright]] | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | year=2008 | isbn=978-0-19-921986-5| page=6 }}</ref> In addition to <math>\gcd(a, b) = 1\;</math> and <math>(a, b) = 1,\;</math> the notation <math>a\perp b</math> is sometimes used to indicate that ''a'' and ''b'' are relatively prime.<ref>{{citation|first1=R. L.|last1=Graham|first2=D. E.|last2=Knuth|first3=O.|last3=Patashnik|title=Concrete Mathematics|publisher=Addison-Wesley|year=1989}}</ref>
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| For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. The numbers 1 and −1 are coprime to every integer, and they are the only integers to be coprime with 0.
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| A fast way to determine whether two numbers are coprime is given by the [[Euclidean algorithm]].
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| The number of integers coprime to a positive integer ''n'', between 1 and ''n'', is given by [[Euler's totient function]] (or Euler's phi function) ''φ''(''n'').
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| A [[Set (mathematics)|set]] of integers can also be called '''coprime''' if its elements share no common positive factor except 1. A set of integers is said to be '''pairwise coprime''' if ''a'' and ''b'' are coprime for every pair (''a'', ''b'') of different integers in it.
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| == Properties ==
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| [[Image:coprime-lattice.svg|thumb|right|300px|Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 x 9 lattice does not intersect any other [[square lattice|lattice points]]]]
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| There are a number of conditions which are equivalent to ''a'' and ''b'' being coprime:
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| *No [[prime number]] divides both ''a'' and ''b''.
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| *There exist integers ''x'' and ''y'' such that ''ax'' + ''by'' = 1 (see [[Bézout's identity]]).
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| *The integer ''b'' has a [[modular multiplicative inverse|multiplicative inverse]] modulo ''a'': there exists an integer ''y'' such that ''by'' ≡ 1 (mod ''a''). In other words, ''b'' is a [[unit (ring theory)|unit]] in the [[ring (mathematics)|ring]] '''Z'''/''a'''''Z''' of [[modular arithmetic|integers modulo]] ''a''.
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| *Every pair of [[congruence relation]]s for an unknown integer ''x'', of the form ''x'' ≡ ''k'' (mod ''a'') and ''x'' ≡ ''l'' (mod ''b''), has a solution, as stated by the [[Chinese remainder theorem]]; in fact the solutions are described by a single congruence relation modulo ''ab''.
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| As a consequence of the third point, if ''a'' and ''b'' are coprime and ''br'' ≡ ''bs'' ([[modular arithmetic|mod]] ''a''), then ''r'' ≡ ''s'' (mod ''a'') (because we may "divide by ''b''" when working modulo ''a''). Furthermore, if ''b''<sub>1</sub> and ''b''<sub>2</sub> are both coprime with ''a'', then so is their product ''b''<sub>1</sub>''b''<sub>2</sub> (modulo ''a'' it is a product of invertible elements, and therefore invertible); this also follows from the first point by [[Euclid's lemma]], which states that if a prime number ''p'' divides a product ''bc'', then ''p'' divides at least one of the factors ''b'', ''c''.
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| As a consequence of the first point, if ''a'' and ''b'' are coprime, then so are any powers ''a''<sup>''k''</sup> and ''b''<sup>''l''</sup>.
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| If ''a'' and ''b'' are coprime and ''a'' divides the product ''bc'', then ''a'' divides ''c''. This can be viewed as a generalization of Euclid's lemma.
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| The two integers ''a'' and ''b'' are coprime if and only if the point with coordinates (''a'', ''b'') in a [[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''). (See figure 1.)
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| In a sense that can be made precise, the [[probability]] that two randomly chosen integers are coprime is 6/π<sup>2</sup> (see [[pi]]), which is about 61%. See below.
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| Two [[natural number]]s ''a'' and ''b'' are coprime if and only if the numbers 2<sup>''a''</sup> − 1 and 2<sup>''b''</sup> − 1 are coprime. As a generalization of this, following easily from [[Euclidean algorithm]] in [[Radix|base]] ''n'' > 1:
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| : <math>\gcd (n^a-1,n^b-1)=n^{\gcd(a,b)}-1.</math>
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| == Coprimality in sets ==
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| A [[Set (mathematics)|set]] of integers ''S'' = {''a''<sub>1</sub>, ''a''<sub>2</sub>, .... ''a''<sub>''n''</sub>} can also be called ''coprime'' or ''setwise coprime'' if the [[greatest common divisor]] of all the elements of the set is 1. If every pair in a (finite or [[infinite set|infinite]]) set of integers is coprime, then the set is said to be ''pairwise coprime'' (or ''pairwise relatively prime'', ''mutually coprime'' or ''mutually relatively prime''). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. For example, the integers 6, 10, 15 are coprime (because the only positive integer dividing ''all'' of them is 1), but they are not ''pairwise'' coprime because the gcd(6, 10) = 2, gcd(10, 15) = 5 and gcd(6, 15) = 3.
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| The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as the [[Chinese remainder theorem]].
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| === Infinite set examples ===
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| The set of all primes is pairwise coprime, as is the set of elements in [[Sylvester's sequence]], and the set of all [[Fermat numbers]].
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| == Coprimality in ring ideals ==
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| Two [[ring ideal|ideals]] ''A'' and ''B'' in the [[commutative ring]] ''R'' are called '''coprime''' (or '''comaximal''') if ''A'' + ''B'' = ''R''. This generalizes [[Bézout's identity]]: with this definition, two [[principal ideal]]s (''a'') and (''b'') in the ring of integers '''Z''' are coprime if and only if ''a'' and ''b'' are coprime. If the ideals ''A'' and ''B'' of ''R'' are coprime, then ''AB'' = ''A''∩''B''; furthermore, if ''C'' is a third ideal such that ''A'' contains ''BC'', then ''A'' contains ''C''. The [[Chinese remainder theorem]] is an important statement about coprime ideals.
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| ==Cross notation, group==
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| {{See also|multiplicative group of integers modulo n}}
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| If ''n''≥1 and is an [[integer]], the numbers coprime to ''n'', taken [[modular arithmetic|modulo]] ''n'', form a [[group (mathematics)|group]] with multiplication as operation; it is written as ('''Z'''/''n'''''Z''')<sup>×</sup> or '''Z'''<sub>n</sub><sup>*</sup>.
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| ==Probabilities==
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| Given two randomly chosen integers ''a'' and ''b'', it is reasonable to ask how likely it is that ''a'' and ''b'' are coprime. In this determination, it is convenient to use the characterization that ''a'' and ''b'' are coprime if and only if no prime number divides both of them (see [[Fundamental theorem of arithmetic]]).
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| Informally, the probability that any number is divisible by a prime (or in fact any integer) <math>p</math> is <math>1/p</math>; for example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by ''p'' is <math>1/p^2</math>, and the probability that at least one of them is not is <math>1-1/p^2</math>. Any finite collection of divisibility events associated to distinct primes is mutually independent. For example, in the case of two events, a number is divisible by primes ''p'' and ''q'' if and only if it is divisible by ''pq''; the latter event has probability 1/''pq''. If one makes the heuristic assumption that such reasoning can be extended to infinitely many divisibility events, one is led to guess that the probability that two numbers are coprime is given by a product over all primes,
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| : <math>\prod_{\text{prime } p} \left(1-\frac{1}{p^2}\right) = \left( \prod_{\text{prime } p} \frac{1}{1-p^{-2}} \right)^{-1} = \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.607927102 \approx 61\%.</math> | |
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| Here ''ζ'' refers to the [[Riemann zeta function]], the identity relating the product over primes to ''ζ''(2) is an example of an [[Euler product]], and the evaluation of ''ζ''(2) as ''π''<sup>2</sup>/6 is the [[Basel problem]], solved by [[Leonhard Euler]] in 1735.
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| There is no way to choose a positive integer at random so that each positive integer occurs with equal probability, but statements about "randomly chosen integers" such as the ones above can be formalized by using the notion of ''[[natural density]]''. For each positive integer ''N'', let ''P''<sub>''N''</sub> be the probability that two randomly chosen numbers in <math>\{1,2,\ldots,N\}</math> are coprime. Although ''P''<sub>''N''</sub> will never equal <math>6/\pi^2</math> exactly, with work<ref>This theorem was proved by [[Ernesto Cesàro]] in 1881. For a proof, see {{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=[[E. M. Wright]] | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | year=2008 | isbn=978-0-19-921986-5}}, theorem 332.</ref> one can show that in the limit as <math>N \to \infty</math>, the probability <math>P_N</math> approaches <math>6/\pi^2</math>.
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| More generally, the probability of ''k'' randomly chosen integers being coprime is 1/''ζ''(''k'').
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| ==Generating all coprime pairs==
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| [[File:Coprime8.svg|300px|thumb|The order of generation of coprime pairs by this algorithm. First node (2,1) is marked red, its three children are shown in orange, third generation is yellow, and so on in the rainbow order.]]
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| All pairs of positive coprime numbers <math>(m, n)</math> (with <math>m > n</math>) can be arranged in two disjoint complete [[ternary tree]]s, one tree starting from <math>(2,1)</math> (for even-odd and odd-even pairs),<ref>{{Citation |last=Saunders |first=Robert |lastauthoramp=yes |last2=Randall |first2=Trevor |title=The family tree of the Pythagorean triplets revisited |journal=Mathematical Gazette |volume=78 |month=July |year=1994 |pages=190–193 }}.</ref> and the other tree starting from <math>(3,1)</math> (for odd-odd pairs).<ref>{{Citation |last=Mitchell |first=Douglas W. |title=An alternative characterisation of all primitive Pythagorean triples |journal=Mathematical Gazette |volume=85 |month=July |year=2001 |pages= 273–275 }}.</ref> The children of each vertex <math>(m,n)</math> are generated as follows:
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| Branch 1: <math>(2m-n,m)</math>
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| Branch 2: <math>(2m+n,m)</math>
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| Branch 3: <math>(m+2n,n)</math>
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| This scheme is exhaustive and non-redundant with no invalid members.
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| ==See also==
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| *[[Superpartient number]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{Citation |last=Lord |first=Nick |title=A uniform construction of some infinite coprime sequences |journal=Mathematical Gazette |volume=92 |month=March |year=2008 |pages=66–70 }}.
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| [[Category:Number theory]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Some time ago he chose to reside in North Carolina and he doesn't strategy on changing it. To climb is something I truly appreciate doing. Invoicing is my occupation.
Feel free to visit my homepage: online psychics (http://youulike.com/)