Coprime integers

Two natural numbers aand bare divisor prime ( a \perp b) if there is no natural number other than one that divides both numbers. Synonymous is relatively prime, from English relatively prime or coprime. If two natural numbers do not have a common prime factor, they are divisorally prime. From this definition it follows that every natural number is prime to 1, including the number 1 itself. Consequently, a fraction of two numbers that are not prime cannot be shortened.

To prove divisor strangeness, one usually calculates the greatest common divisor: Two numbers are prime if 1 is their greatest common divisor.

More than two natural numbers are called pairwise coprime if any two of them are mutually divisible, and divisible if there is no prime factor that all these numbers have in common. Numbers which are pairwise coprime are also always divisible. The reverse conclusion does not hold, because, for example, 6, 10, 15 are divisible but not pairwise divisible (e.g., because ggT(10, 15) = 5).

Examples

  • The numbers 12 and 77 are alien to each other, because their prime factorizations 12 = 2 - 2 - 3 and 77 = 7 - 11 do not contain any common prime factors.
  • The numbers 15 and 25 are not alien to each other, because in their prime factorizations 15 = 3 - 5 and 25 = 5 - 5 the 5 occurs, which is also the ggT(15, 25).
  • The numbers 9, 17, 64 are pairs of divisors, because all three pairs 9 and 17, 17 and 64, 9 and 64 are pairs of divisors.

Obviously, two distinct primes are always prime, since they have only themselves as prime factor. Other examples of divergent numbers are two numbers whose difference is 1, or two odd numbers whose difference is 2.

Divisor strangeness occurs, often as a condition, in many number theoretic problems. For example, a condition for the ChineseRemainder Theorem is that the moduli are divisor alien. The Eulerian φ-function assigns to each natural number n the number of numbers that are divisor alien to n in . \{1,\dots ,n\}

Properties

Divider strangeness is a binary relation

{\displaystyle {\text{Teilerfremdheit}}=\left\{\left(a,b\right)\in \mathbb {N} \times \mathbb {N} \ \vert \ \operatorname {ggT} (a,b)=1\right\}}

This relation is not transitive, because, for example, 2 and 3 are alien to each other, as are 3 and 4, but not 2 and 4.

The asymptotic probability that two randomly chosen integers aand bare divisorally distinct is

{\displaystyle P(\operatorname {ggT} (a,b)=1)={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\,\%,}

where is \zeta the Riemann ζ-function and π \pi is the circle number. This theorem was first proved by Ernesto Cesàro in 1881.

general, is {\displaystyle 1/(r^{n}\,\zeta (n))}the asymptotic density of n-tuples with greatest common divisor r.


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