A starting point could be to first solve the recurrence relation:

[LaTeX ERROR: Convert failed]

So [LaTeX ERROR: Convert failed]

Then [LaTeX ERROR: Convert failed]

In general [LaTeX ERROR: Convert failed] where we've used the formula for the geometric sum. Now after modding out m, we need to be left with m distinct numbers

To show that they are all different, I think you need to use something of the sort: [LaTeX ERROR: Convert failed]