By calculating directly show that the number of solutions to is if and if . (Hint: Use the change of variables and .)
- Case
First, note that the mapping is invertible. Further if then and are coprime to since we have
Let be a primitive root module ( refers to a prime), then for some s and t with since we want the incongruent solutions, so here note that for each you have a unique such that that happens. So we have p-1 incongruent solutions module
Now you may check easily that the solutions are incongruent mod p if and only if the correspoding solutions are incongruent (with ) hence we have incogruent solutions.
- Case
(because is prime)
If then we must have
Otherwsie, for each we have and as incongruent solutions. (again we must have )
So we have: solutions
Just in case, I say that 2 solutions to our equation and are incongruent mod. p if and only if and do not hold simultaneously