modular arithmetic problem

[JuanTutors]

I can't remember one of the things I learned in algebra class:

Given odd numbers N and r, suppose we know n is an rth root of k. Are there any general or special cases when we can find a different rth root of k?

[R.D. Silverman]

Originally Posted by dominicanpapi82

I can't remember one of the things I learned in algebra class:

Given odd numbers N and r, suppose we know n is an rth root of k. Are there any general or special cases when we can find a different rth root of k?

Huh? The question makes ZERO sense.

(1) You have not specified the domain in which you are working.
(2) If you are working in the reals, only one such rth root ever exists.
(3) If you are working over C, then k always has r rth roots. What do
you mean by "special cases when we can find a different rth root"?
What do you mean by "find"??? Determining all of the r'th roots of k
is trivially given by DeMoivre's Theorem. This is high school level math.
(4) Your notation sucks. You use both N and n to mean the same thing.
Or, if you intend that they denote different numbers, then you have not defined N.

[JuanTutors]

Of course. Sorry, I didn't realize I didn't type mod N.

Restated correctly: Given odd numbers N and r, suppose we know n is an rth root of k mod N. Are there any general or special cases when we can find a different rth root of k mod N?

I think I know the answer to the question, but I wanted to verify it.

Originally Posted by R.D. Silverman

Huh? The question makes ZERO sense.

(1) You have not specified the domain in which you are working.
(2) If you are working in the reals, only one such rth root ever exists.
(3) If you are working over C, then k always has r rth roots. What do
you mean by "special cases when we can find a different rth root"?
What do you mean by "find"??? Determining all of the r'th roots of k
is trivially given by DeMoivre's Theorem. This is high school level math.
(4) Your notation sucks. You use both N and n to mean the same thing.
Or, if you intend that they denote different numbers, then you have not defined N.

[R.D. Silverman]

Originally Posted by dominicanpapi82

Of course. Sorry, I didn't realize I didn't type mod N.

Restated correctly: Given odd numbers N and r, suppose we know n is an rth root of k mod N. Are there any general or special cases when we can find a different rth root of k mod N?

I think I know the answer to the question, but I wanted to verify it.

What do you mean by "find".

Sometimes, (many times) when r is odd, there is only one such root.

When there exists more than one, there are a variety of methods
to "find" them; e.g. Berlekamp, Cantor-Zassenhaus, variations of
Shanks-Tonelli, etc.

[maxal]

Originally Posted by dominicanpapi82

Given odd numbers N and r, suppose we know n is an rth root of k mod N. Are there any general or special cases when we can find a different rth root of k mod N?

Just multiply n by any r-th power root of 1 modulo N. In particular, if r is even, you can multiply n by -1 to get a different r-th power root of k mod N.