What are examples of coprime numbers

Coprime integers - Coprime integers

A collection of numbers whose only common factor is 1
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In number theory, two whole numbers a and b are coprime , relatively prim or mutually indivisible if the only positive integer that will divide equally (is a divisor of) both of them are 1. One also says a prim b or a is with coprime b . Hence, any prime that divides one of a or b does not divide the other. This corresponds to the greatest common divisor (gcd) of 1.

The numerator and denominator of a reduced fraction are coprime. The numbers 14 and 25 are coprime since 1 is their only common factor. On the other hand, 14 and 21 are not coprimes because they are both divisible by 7.

Notation and examination

Standard notations for relatively prime numbers a and b are: gcd ( a , b ) = 1 and ( a , b ) = 1. In a 1989 paper, Graham, Knuth, and Patashnik suggested using the notation to indicate that a and b are relatively prime and that the term "prime" is used in place of coprime (as is in a prim to b).

A quick way to determine whether two numbers are coprime is offered by the Euclidean algorithm and its faster variants such as the binary GCD algorithm or the Lehmer GCD algorithm.

The number of integers starting with a positive integer n between 1 and n coprime is given by the Euler totient function, also called Euler's phi function φ ( n ) is known.

A lot of integers can also be used as Are called coprime, if their elements have no common positive factor other than 1. A stronger condition for a set of integers is that in pairs coprime , which means, that a and b Coprime for each couple ( a , b ) of different are integers in the set. The set {2, 3, 4} is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.

properties

The numbers 1 and -1 are the only integers coprime with any integer, and they are the only integers coprime with 0.

Matches a number of conditions a and b as coprime:

As a consequence of the third point, if a and b Coprime and br bs (mod a ) are, then is r s (mod a ). That means we can "go through." b share "when we modulo a work . If b 1 and b 2 both with a are coprime, this is their product too b 1 b 2 (i.e. modulo a is a product of invertible elements and therefore invertible); This also follows from the first point of Euclid's lemma, which says that if is a prime number p a product bc Splits , p at least one of the factors b , c Splits .

As a result of the first point, if a and b Coprime are, too, all potencies a k and b m .

If a and b are coprime and a shares the product bc , then a to divide c . This can be seen as a generalization of Euclid's lemma.

Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 × 9 grid does not intersect other grid points

The two whole numbers a and b are coprime if and only if the point with the coordinates ( a , b ) is "visible" in a Cartesian coordinate system from the origin (0,0), in the sense that there is no point with integer coordinates on the line segment between the origin and ( a , b ). (See Figure 1.)

In a precise sense, the probability that two randomly chosen integers are coprimes is 6 / π 2 (see pi), which corresponds to approximately 61%. See below.

Two natural numbers a and b are coprime if and only if the numbers 2 a - 1 and 2 b - 1 coprime are. As a generalization, it follows from the Euclidean algorithm in Basis n  > 1:

Coprimality in quantities

A set of whole numbers S. = { a 1 , a 2 , .... a n } can also be called coprime or setwise coprime For example, if the greatest common divisor of all the elements of the set is 1 then the integers 6, 10, 15 are coprimes because 1 is the only positive integer that divides all of them.

If each pair in a set of integers is coprime, the set is called labeled coprime in pairs (or relatively prime in pairs , mutually coprime or each other relatively prim ). The pairwise coprimality is a stronger condition than the sentence coprimality. Any pairwise coprime finite set is also clause-wise coprime, but the opposite is not the case. For example, the integers 4, 5, 6 (sentence-wise) coprime (because the only positive integer they all divides, 1 is), but they are not in pairs Koprime (because gcd (4, 6) = 2).

The concept of pairwise coprimality is important as a hypothesis in many results of number theory, such as in the Chinese remainder theorem.

It is possible for an infinite set of integers to be pairwise coprime. Notable examples are the set of all prime numbers, the set of elements in Sylvester's sequence, and the set of all Fermat numbers.

Coprimality in ring ideals

Two ideals A. and B. in a commutative ring R. called coprime (or comaximal ) , if A. + B. = R. . This generalizes the identity of Bézout: with this definition, two main ideals ( a ) and ( b ) in the ring of whole numbers Z coprime if and only if a and b Are coprime. If the ideals A. and B. of R. Are coprime, then is FROM = A. B. ; Besides, if C. a third is ideal so that A. contains BC , then A. contains C. . The Chinese remainder theorem can be generalized to any commutative ring using Coprime ideals.

Probability of coprimality

With two randomly selected whole numbers a and b is it reasonable to ask how likely it is that a and b Are coprime. In making this determination it is appropriate to use the characterization that a and b are coprimes if and only if no prime number divides both (see fundamental theorem of arithmetic).

Informal is the probability that a number is divisible by a prime number (or actually an integer); For example, every 7th whole number is divisible by 7. Hence the probability that two numbers are both through p are divisible, and the probability that at least one of them is not divisible is the same. Every finite collection of divisibility events assigned to certain prime numbers is independent of one another. For example, in the case of two events, a number is through the prime numbers if and only then p and q divisible if by pq is divisible; the latter event has a probability of 1 / pq . If one makes the heuristic assumption that such an argument can be extended to infinitely many divisibility events, one is led to the conjecture that the probability that two numbers are coprimes is given by a product over all prime numbers.

Here ζ This is what Riemann refers to as the zeta function that relates the identity of the product via prime numbers ζ (2) is an example of a Euler product, and the evaluation of the ζ (2) as π 2 / 6 the Basel problem was solved by Leonhard Euler in 1735.

There is no way to randomly select a positive integer so that every positive integer has an equal probability of occurrence, but statements about "random integers" such as those mentioned above can be made using the term of natural density be formalized. For any positive integer N be P N the probability that two randomly selected numbers are in coprime. Although P N will never be exactly the same, one can show with the work that the probability approaches in the borderline case.

More general is the probability that k randomly selected integers are coprimes.

Generate all coprime pairs

The order in which this algorithm creates coprime pairs. The first node (2,1) is marked in red, its three children are shown in orange, the third generation is yellow and so on in rainbow order.

All pairs of positive coprime numbers (with) can be arranged in two disjoint complete ternary trees, with one tree starting from (for even - odd and odd - even pairs) and the other tree starting from (for odd - odd pairs). The children of each vertex are generated as follows:

  • Branch 1:
  • Branch 2:
  • Branch 3:

This schema is complete and non-redundant with no invalid members.

Applications

When designing the machine, even gearwheel wear is achieved by choosing the number of teeth of the two interlocking gearwheels to be relatively prime. If a gear ratio of 1: 1 is desired, a gear relative to the two equally sized gears can be inserted between them.

In pre-computer cryptography, some Vernam cipher machines combined multiple loops of lanyards of different lengths. Many rotor machines combine rotors with different numbers of teeth. Such combinations work best when the entire set of lengths is pairwise coprime.

See also

Remarks

References

  • Hardy, GH; Wright, EM (2008), An introduction to number theory (6th ed.), Oxford University Press, ISBN
  • Niven, Ivan; Zuckerman, Herbert S. (1966), An introduction to number theory (2nd ed.), John Wiley & Sons
  • Ore, Oystein (1988) [1948], Number theory and its history , Dover, ISBN
  • Rosen, Kenneth H. (1992), Elementary number theory and its applications (3rd ed.), Addison-Wesley, ISBN

further reading

  • Lord, Nick (Mar 2008), "A Uniform Construction of Some Infinite Coprime Sequences", Mathematical Gazette , 92 : 66–70 .