text: # Thread: find primitive roots for mod
  
  1. ## find primitive roots for mod
  
  is there an easy way to find primitive roots for a specific mod?
  
  2. There isn't a specific formula to find primitive roots. However, say that you have found one primitive root, $\displaystyle g$ modulo $\displaystyle m$. Then, $\displaystyle g^k$ is also a primitive root iff $\displaystyle (\varphi (m), k) = 1$ and this gives the complete list.
  
  More here: How to find a primitive root of N
  
  3. So say I have found a primitive root mod 29. namely 2.
  
  then the other primitive roots mod 29 are powers of 2?
  
  that means that 8 is also a primitive root because 2^3=8 and (28,3)=1.
  
  but 2^5=32 (28,5)=1 does that mean that 3 is a primitive root because 32 mod 29 is congruent to 3?
  
  is the only other primitive root for 29 8?
  
  4. Yes. $\displaystyle 2^k$ is a primitive root mod 29 iff $\displaystyle (k,28) = 1$.
  
  So for example, $\displaystyle 2^3 \equiv 8$ is a primitive root since $\displaystyle (3, 28)=1$
  
  5. but that is the only other p.r. besides 2. correct?
  
  6. No ... there are many other values for $\displaystyle k$ such that $\displaystyle (k,28) = 1$.
  
  Try $\displaystyle k = 1, 3, 5, 7, 9, 11, 13, ...$
  
  In general, every prime $\displaystyle p$ has $\displaystyle \varphi (p-1)$ primitive roots.
explanation: The document is a thread from a forum discussing the topic of finding primitive roots modulo a specific number. It contains a series of questions and answers about the methods and properties of primitive roots, particularly focusing on the example of modulo 29.
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distilabel_metadata.content: # Thread: find primitive roots for mod
  
  1. ## find primitive roots for mod
  
  is there an easy way to find primitive roots for a specific mod?
  
  2. There isn't a specific formula to find primitive roots. However, say that you have found one primitive root, $\displaystyle g$ modulo $\displaystyle m$. Then, $\displaystyle g^k$ is also a primitive root iff $\displaystyle (\varphi (m), k) = 1$ and this gives the complete list.
  
  More here: How to find a primitive root of N
  
  3. So say I have found a primitive root mod 29. namely 2.
  
  then the other primitive roots mod 29 are powers of 2?
  
  that means that 8 is also a primitive root because 2^3=8 and (28,3)=1.
  
  but 2^5=32 (28,5)=1 does that mean that 3 is a primitive root because 32 mod 29 is congruent to 3?
  
  is the only other primitive root for 29 8?
  
  4. Yes. $\displaystyle 2^k$ is a primitive root mod 29 iff $\displaystyle (k,28) = 1$.
  
  So for example, $\displaystyle 2^3 \equiv 8$ is a primitive root since $\displaystyle (3, 28)=1$
  
  5. but that is the only other p.r. besides 2. correct?
  
  6. No ... there are many other values for $\displaystyle k$ such that $\displaystyle (k,28) = 1$.
  
  Try $\displaystyle k = 1, 3, 5, 7, 9, 11, 13, ...$
  
  In general, every prime $\displaystyle p$ has $\displaystyle \varphi (p-1)$ primitive roots.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a thread from a forum discussing the topic of finding primitive roots modulo a specific number. It contains a series of questions and answers about the methods and properties of primitive roots, particularly focusing on the example of modulo 29.
  
  Label: multiple-question-and-answer