Chapter 12

Exercise 1. This software finds all solutions to the equation x² + y² = 0 in Z_p.  Run your program for all odd primes up to 37. Make a conjecture about the the number of solutions in Z_p (where p is a prime) and the form of p.

Exercise 4. This software determines the order of the group of units in the ring of 2 by 2 matrices over Z_n (that is, the group GL(2,Z_n)) and the subgroup SL(2,Z_n). Run the program for n = 2, 3, 5, 7, 11, and 13. What relationship do you see between the order of GL(2,Z_n) and the order of SL(2,Z_n) in these cases? Run the program for n = 16, 27, 25, and 49. Make a conjecture about the relationship between the order of GL(2,Z_n) and the order of SL(2,Z_n) when n is a power of a prime. Run the program for n = 32. (Notice that when you run the program for n = 32 the table shows the orders for all divisors of 32 greater than 1.) How do the orders the two groups change each time you increase the power of 2 by 1? Run the program for n = 27. How do the orders the two groups change each time you increase the power of 3 by 1? Run the program for n = 25. How do the orders the two groups change when you increase the power of 5 by 1? Make a conjecture about the relationship between |SL(2,Z_pⁱ)| and |SL(2,Z_pⁱ⁺¹)|. Make a conjecture about the relationship between |GL(2,Z_pⁱ)| and |GL(2,Z_pⁱ⁺¹)|. Run the program for n = 12, 15, 20, 21, and 30. Make a conjecture about the order of GL(2,Z_n) in terms of the orders of GL(2,Z_s) and GL(2,Z_t) where n = st and s and t are relatively prime. (Notice that when you run the program for st the table shows the values for st, s and t .) For each value of n is the order of SL(2,Z_n) divisible by n? Is it divisible by n + 1? Is it divisible by n - 1?

 

Exercise 5. In the ring Z_n this software finds the number of solutions to the equation x² = -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?

 

Exercise 6. This software determines the number of solutions to the equation X² = -I where X is a 2 x 2 matrix with entries from Z_n and I is the identity. Run the program for n = 32. Make a conjecture about the number of solutions when n = 2^k where k > 1. Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3. Run the program for n = 27 and 49. Make a conjecture about the number of solutions when n has the form pⁱ where p is a prime of the form 4q + 3. Run the program for n = 5, 13, 17, 29, and 37. Make a conjecture about the number of solutions when n is a prime of the form 4q + 1. Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n ?