Proving Primitive Roots of Odd Numbers Modulo pm

[Gear300]

Hello friends from afar.

I ran into what I felt to be somewhat of an odd question:

Prove that some odd numbers are primitive roots modulo p^m for each odd prime p and each positive integer m.

It feels dodgy given that any odd number n = p₁p₂ ⋅⋅⋅ p_s cannot be a primitive root of a prime number involved in its prime factorization. I just needed to be sure. Many thanks.

 

[fresh_42]

The wording is quite disturbing and I stumbled upon the same argument as you. "some odd numbers" looks strange.
It would make more sense the other way around (or I didn't get the point either):

For each odd prime $p$ and each positive integer $m$ prove that some odd numbers are primitive roots modulo $p^m.$

 

[Gear300]

Indeed. I'm guessing yours is how it's done, since it seems like the original could be semantically interpreted like that. Thanks.

 

[Mark44]

Hello friends from afar.

I ran into what I felt to be somewhat of an odd question:

Prove that some odd numbers are primitive roots modulo p^m for each odd prime p and each positive integer m.

It feels dodgy given that any odd number n = p₁p₂ ⋅⋅⋅ p_s cannot be a primitive root of a prime number involved in its prime factorization. I just needed to be sure. Many thanks.

In future posts, please don't delete the homework template...