Exercise 5. In the ring $$Z_n$$ this software finds the number of solutions to the equation $$x^2 = -1$$. Run the program for all primes between $$3$$ and $$29$$. How does the answer depend on the prime? Make a conjecture about the number of solutions when $$n$$ is a prime greater than $$2$$. Run the program for the squares of all primes between $$3$$ and $$29$$. Make a conjecture about the number of solutions when $$n$$ is the square of a prime greater than $$2$$. Run the program for the cubes of primes between $$3$$ and $$29$$. Make a conjecture about the number of solutions when $$n$$ is any power of an odd prime. Run the program for $$n = 2, 4, 8, 16$$, and $$32$$. Make a conjecture about the number of solutions when $$n$$ is a power of $$2$$. Run the program for $$n = 12, 20, 24, 28$$, and $$36$$. Make a conjecture about the number of solutions when $$n$$ is a multiple of $$4$$. Run the program for various cases where $$n = pq$$ and $$n = 2pq$$ where $$p$$ and $$q$$ are odd primes. Make a conjecture about the number of solutions when $$n = pq$$ or $$n = 2pq$$ where $$p$$ and $$q$$ are odd primes. What relationship do you see between the number of solutions for $$n = p$$ and $$n = q$$ and $$n = pq$$? Run the program for various cases where $$n = pqr$$ and $$n = 2pqr$$ where $$p, q$$ and $$r$$ are odd primes. Make a conjecture about the number of solutions when $$n = pqr$$ or $$n = 2pqr$$ where $$p$$, $$q$$ and $$r$$ are odd primes. What relationship do you see between the number of solutions when $$n = p$$, $$n = q$$ and $$n = r$$ and the case that $$n = pqr$$?

Please enter $$n$$ and click the button, the solutions will show below.