Information about Coprime
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 number 1 is coprime to every integer.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n which are coprime to n.
There are a number of conditions which are equivalent to a and b being coprime:
If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c.
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.)
The probability that two randomly chosen integers are coprime is 6/π2 (see pi), which is about 60%. See below.
Two natural numbers a and b are coprime if and only if the numbers 2a − 1 and 2b − 1 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.
The concept of being relatively prime can also be extended any finite set of integers S = {a1, a2, .... an} to mean that the greatest common divisor of the elements of the set is 1. If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime.
Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor.)
and 
, it is reasonable to ask how likely it is that and are coprime. In this determination, it is convenient to use the characterization that and are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic).
The probability that any number is divisible by a prime (or any integer),
is 
. Hence the probability that two numbers are both divisible by this prime is 
, and the probability that at least one of them is not is 
. Thus the probability that two numbers are coprime is given by a product over all primes,
Here
refers to the Riemann zeta function. In general, the probability of 
randomly chosen integers being coprime is 
.
There is often confusion about what a "randomly chosen integer" is. One way of understanding this is to assume that the integers are chosen randomly between 1 and an integer
. Then for each upper bound , there is a probability 
that two randomly chosen numbers are coprime. This will never be exactly 
, but in the limit as 
, 
.
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For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. The number 1 is coprime to every integer.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n which are coprime to n.
Properties
)
Figure 1. The numbers 4 and 9 are coprime because the diagonal does not intersect any other lattice points
There are a number of conditions which are equivalent to a and b being coprime:
- There exist integers x and y such that ax + by = 1 (see Bézout's identity).
- The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZ of integers modulo a.
If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c.
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.)
The probability that two randomly chosen integers are coprime is 6/π2 (see pi), which is about 60%. See below.
Two natural numbers a and b are coprime if and only if the numbers 2a − 1 and 2b − 1 are coprime.
Cross notation, group
If n≥1 is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)× or Zn*.Generalizations
Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (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.
The concept of being relatively prime can also be extended any finite set of integers S = {a1, a2, .... an} to mean that the greatest common divisor of the elements of the set is 1. If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime.
Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor.)
Probabilities
Given two randomly chosen integers and , it is reasonable to ask how likely it is that and are coprime. In this determination, it is convenient to use the characterization that and are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic).
The probability that any number is divisible by a prime (or any integer),
is . Hence the probability that two numbers are both divisible by this prime is , and the probability that at least one of them is not is . Thus the probability that two numbers are coprime is given by a product over all primes,
Here
refers to the Riemann zeta function. In general, the probability of randomly chosen integers being coprime is .
There is often confusion about what a "randomly chosen integer" is. One way of understanding this is to assume that the integers are chosen randomly between 1 and an integer
. Then for each upper bound , there is a probability that two randomly chosen numbers are coprime. This will never be exactly , but in the limit as , .
See also
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
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divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.
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Explanation
For example, 7 is a divisor of 42 because 42/7 = 6...... Click the link for more information.
In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder.
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6 (six) is the natural number following 5 and preceding 7.
The SI prefix for 10006 is exa (E), and for its reciprocal atto (a).
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The SI prefix for 10006 is exa (E), and for its reciprocal atto (a).
In mathematics
Six is the second smallest composite number, its proper divisors being 1, 2 and 3...... Click the link for more information.
35 (thirty-five) is the natural number following 34 and preceding 36.
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In mathematics
It is the sum of the first five triangular numbers, making it a tetrahedral number. It is also a centered cube number. 35 is a pentagonal number and a pentatope number...... Click the link for more information.
27 (twenty-seven) is the natural number following 26 and preceding 28. Twenty-seven is the smallest positive integer requiring four syllables to name in English, though it can be unambiguously defined in just two: "three cubed.
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This article is about the number one. For the year AD 1, see 1. For other uses, see 1 (disambiguation).
0 1 2 3 4 5 6 7 8 9 →
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Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers).
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totient of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n. For example, since the six numbers 1, 2, 4, 5, 7 and 8 are coprime to 9. The function so defined is the totient function.
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In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation. It states that if a and b are nonzero integers with greatest common divisor d, then there exist integers x and y (called
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multiplicative inverse for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
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In mathematics, a unit in a (unital) ring R is an invertible element of R, i.e. an element u such that there is a v in R with
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- uv = vu = 1R, where 1R
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
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Euclid's lemma (Greek λῆμμα) is a generalization of Proposition 30 of Book VII of Euclid's Elements.
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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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Probability is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of
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In mathematics, a natural number can mean either an element of the set (i.e the positive integers or the counting numbers) or an element of the set (i.e. the non-negative integers).
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus.
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group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
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In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation. It states that if a and b are nonzero integers with greatest common divisor d, then there exist integers x and y (called
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In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.
More specifically:
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More specifically:
- a left principal ideal of R is a subset of R
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