text: # find primitive roots for mod
  
  • Feb 17th 2009, 01:50 PM
  mathisthebestpuzzle
  find primitive roots for mod
  is there an easy way to find primitive roots for a specific mod?
  • Feb 17th 2009, 02:25 PM
  o_O
  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
  • Feb 17th 2009, 02:31 PM
  mathisthebestpuzzle
  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?
  • Feb 17th 2009, 02:51 PM
  o_O
  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 $\displaystyle (3, 28)=1$
  • Feb 17th 2009, 03:03 PM
  mathisthebestpuzzle
  but that is the only other p.r. besides 2. correct?
  • Feb 17th 2009, 03:55 PM
  o_O
  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.