Coprime numbers

In mathematics, coprime numbers (numbers that are prime to each other or relatively primes) are two integers a and b that have no prime factors in common. In other words, if they have no other common divisor than 1 and -1. They are equivalently coprime, if and only if their greatest common divisor (GCD) is equal to 1. Two coprime numbers do not have to be individually prime.. 14 and 15 are composite, however they are coprime, since their GCD =1.

For example, 6 and 19 are coprime, but 6 and 27 are not because they are both divisible by 3. 1 is coprime with respect to all integers, while 0 is coprime only with respect to 1 and -1.

A quick calculation to determine if two integers are coprime is Euclid's algorithm.

Properties

Basic

• If two whole numbers a and b are cousins to each other, then there are two integers x e and / a·x + b·and = 1. (Bézout identity)

• Yeah. a and b They're coprimos, too. a divide the product bc, then a divide a c. (Lema de Euclides)

• The integers a and b They're coprimos when b has an inverse for the module product a; that is, there is an integer and such as b·and ≡ 1 (mod a). One consequence of this is that if a and b are cousins to each other and bm ≡ bn (mod) a) then m ≡ n (mod) a). In other words, b is simplified in the ring Z/nZ of the entire modules a.
• If natural numbers a and b are coprimos are also to², ab, b²♪
• If the positive integers m and n are coprimos, they are also m, n, m+n..
• If a is integer, a and a+1 are coprimos.

Other properties

Figure 1. Numbers 4 and 9 are coprimos. Therefore, the diagonal of reticle 4 x 9 does not interfere with any of the other points of the reticle.

• The two integers a and b are cousins to each other, yes and only if, the coordinates point (a, b) in a cartesian coordinate system is visible from the origin (0.0) in the sense that there is no whole coordinate point between the origin and (a,b(see figure 1).

• The probability that two randomly chosen integers are cousins to each other is equal to 6/π2.

• Two natural numbers a and b are cousins to each other, yes and only yes, numbers 2^a-1 and 2^b-1 are cousins to each other. As a generalization of this, it is easily followed by the Euclides algorithm based on n/2003/1:

gcd(na− − 1,nb− − 1)=ngcd(a,b)− − 1.{displaystyle gcd(n^{a}-1,n^{b}-1)=n^{gcd(a,b)}-1. !

• Number of minor natural numbers n and are coprimos with it, provided by the function φ of Euler φ(n).

• If two natural numbers are consecutive then they are coprimos (rest = 1, by the Algorithm of Euclides).

Proposition

Every divisor of the sum of two coprime squares is equal to the sum of two squares.

Example

41 divide to 1681 = 9²+40², (1600 and 81 are coprimos) then 41 = 5²+4², sum of squares.

Generalization

Two ideals I and J in a commutative ring A are coprime if I + J = A. This generalizes the Bézout identity. If I and J are prime to each other, then IJ = I∩J; furthermore, if K is a third ideal such that I contains JK, then I contains K.

With this definition, two principal ideals (a) and (b) in the ring of integers Z are prime to each other, if and only if a and b are prime to each other.