show that primitive root modulo 2^t where t is positive integer, does not exist for t>2

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- Apr 28th 2008, 08:49 AMmandy123primitive root
show that primitive root modulo 2^t where t is positive integer, does not exist for t>2

- Apr 28th 2008, 09:35 AMicemanfan
- Apr 28th 2008, 09:36 AMThePerfectHacker
The proof depends on the fact that for . This shows that the order of is .

Once you established that prove that are all incongruent with eachother. Then are all incrongruent with eachother. And finally each element in is incongruenct with each element of .

With those two facts the proof is almost complete. Because there are a total of in and . Which means is a__reduced system of residues__. And each element does not have order . Which means it has no primitive root.