Almost prime

A - almost-prime or almost-prime -order is a natural number whose prime factorization consists of exactly primes, where multiple prime factors are counted as often. Since all natural numbers are composed of prime factors, every natural number is also a fast prime. Fast primes of the second order is also called semi- primes. Fast primes move between the poles of the indivisible prime numbers and the maximum divisible highly composite numbers and close it with both a.

The Norwegian Viggo Brun introduced the term around 1915 for a generalization of prime numbers to find a new approach to prime unsolved problems.

Definition

Be with primes. Then is called almost-prime -order, which applies. The numbers for a solid is also referred to. The well- definedness follows from the uniqueness of the prime factorization for all natural numbers.

This concept can easily be generalized to the whole numbers and any ZPE -rings.

Properties

  • Each prime number is a almost-prime of order 1, every composite number is an almost-prime of order 2 or higher. Fast primes third order, provided they consist of three distinct prime factors, also called sphenische numbers.
  • The union of form a partition of the natural numbers.
  • Each almost-prime -order is the product of almost-primes of the orders with such The product of the 3- prime Fast 12 and the 4 -Fast 40 prime prime gives the 7 -Fast 480, there is such possible for partitions, wherein the Stirling numbers of the second kind referred to.
  • Since there is no possible prime factorization for the zero, it is not almost-prime -order.
  • The submitter is assigned the empty product as prime factorization. Accordingly, they can be compliant definition called almost-prime 0 -th order.

Examples and values

  • Is an almost-prime first-order ( " prime ").
  • Is a second-order almost-prime ( " semi- prime ").
  • Is an almost-prime fourth order.
  • Is an almost-prime tenth order.

Applications

Fast primes of the second order, ie the product of two prime numbers, refer to the cryptography application.

The following two sentences have been proven in the 1970s:

  • Any sufficiently large, natural, even number can be represented as the sum of an almost-prime first and a second almost-prime order. This statement is similar to the Goldbach 's conjecture.
  • There are infinitely many prime numbers, which have a distance of 2 to a 2- almost-prime. This is similar to the twin primes conjecture about.
Integer Sphenic number Hans Riesel Paulo Ribenboim Wolfram Research
327015
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