Find Number of Generators in Z_(p^r)

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Discussion Overview

The discussion focuses on finding the number of generators of the cyclic group Z_(p^r), where p is a prime number and r is an integer greater than or equal to 1. The conversation includes hints and questions related to the properties of generators in cyclic groups.

Discussion Character

  • Exploratory, Technical explanation

Main Points Raised

  • One participant asks for a hint regarding the number of generators of Z_(p^r).
  • Another participant inquires about the general knowledge of generators of cyclic groups.
  • It is proposed that the generators are those integers that are relatively prime to p^r.
  • A later reply suggests that there is an algorithm for computing how many integers in the range [0, n] are relatively prime to any number n.

Areas of Agreement / Disagreement

Participants appear to agree on the notion that generators of Z_(p^r) are related to integers that are relatively prime to p^r, but the discussion does not reach a consensus on the specific method for counting them.

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?
 
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What do you know about generators of cyclic groups?
 
Well, the generators are those which are relatively prime to p^r.
 
Eivind said:
Well, the generators are those which are relatively prime to p^r.
And don't you know an algorithm for computing how many integers in the range [0, n] are relatively prime to any number n?
 

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