Primorial
With primorial (of English. Primorial ) and the Primfakultät is defined as the product of all primes that do not exceed a certain number. The terms are closely related to the faculty and are mainly used in the mathematical field of number theory are used.
The name primorial is the Germanized English word primorial. The product of primes is however rarely mentioned in the German primorial, even rarer Primfakultät less than or equal. Usually it is described as " the product of the primes less than or equal n".
Definition
Is a natural number, the Primfakultät is defined as the product of all prime numbers less than or equal:
Sometimes a distinction is the special case in which prime and defined only for this analogy the primorial that remains undefined for non- prime.
In the case of the empty product is available, the value of Primfakultät and Primorials is then 1 for arguments that are not prime, the primorial has no values . The Primfakultät provides the value would give the next smallest prime number for this. In practical use, however, the two terms are often used as a synonym.
Example
To calculate the value of the Primorials, one first determines all prime numbers less than or equal 7 These are 2, 3, 5 and 7 provides the product of these four primes. For 9 could, however, not primorial, but rather calculate the Primfakultät - because 9 is not prime and the next smallest prime number 7 and the next largest prime number which is 11, is considered.
Properties
- Let and two adjacent primes. Then for any natural number:
- For the primorial you know the following estimate
- In addition:
- The number of divisors of Primorials depends on the function, that is the number # 2 has 2 dividers, 3 # has 4 dividers, 5 # has 8, and 97 # has already divider etc.
- The sum of the reciprocal values of the Primfakultät converges to a constant