- #1
Eivind
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Let p be a prime number. Find the number of generators of the cyclic group Z_(p^r), where r is an integer >=1.
A small hint, please?
A small hint, please?
And don't you know an algorithm for computing how many integers in the range [0, n] are relatively prime to any number n?Eivind said:Well, the generators are those which are relatively prime to p^r.
Z_(p^r) is a mathematical notation that represents the set of integers modulo p^r, where p is a prime number and r is a positive integer. It includes all the numbers from 0 to p^r-1, and follows specific rules for addition and multiplication.
The number of generators in Z_(p^r) is equal to the number of positive integers less than p^r that are relatively prime to p^r. This can be calculated using Euler's totient function, which counts the number of positive integers less than a given number that are coprime to it.
Knowing the number of generators in Z_(p^r) is important in various mathematical applications, such as cryptography and number theory. It can also help us understand the structure and properties of this set, and can be used in solving related problems.
Yes, the number of generators in Z_(p^r) can change depending on the values of p and r. For example, if p is a small prime number, there may be many generators in Z_(p^r), but as p increases or r becomes larger, the number of generators may decrease.
There is no general formula for calculating the number of generators in Z_(p^r), but it can be determined by using various mathematical techniques such as Euler's totient function or the Chinese remainder theorem. The exact method used may depend on the specific values of p and r.