High School Mathematics Extensions/Primes/Project/The Square Root of -1

Project -- The Square Root of -1 Notation: In modular arithmetic, if for some m, then we can write we say, x is the square root of y mod m. Note that if x satisfies x2 ≡ y, then m - x ≡ -x when squared is also equivalent to y. We consider both x and -x to be square roots of y. 1. Question 5 of the Problem Set showed that exists for p ≡ 1 (mod 4) prime. Explain why no square root of -1 exist if p ≡ 3 (mod 4) prime. 2. Show that for p ≡ 1 (mod 4) prime, there are exactly 2 solutions to 3. Suppose m and n are integers with gcd(n,m) = 1. Show that for each of the numbers 0, 1, 2, 3, .... , nm - 1 there is a unique pair of numbers a and b such that the smallest number x that satisfies: x ≡ a (mod m) x ≡ b (mod n) is that number. E.g. Suppose m = 2, n = 3, then 4 is uniquely represented by x ≡ 0 (mod 2) x ≡ 1 (mod 3) as the smallest x that satisfies the above two congruencies is 4. In this case the unique pair of numbers are 0 and 1. 4. If p ≡ 1 (mod 4) prime and q ≡ 3 (mod 4) prime. Does have a solution? Why? 5. If p ≡ 1 (mod 4) prime and q ≡ 1 (mod 4) prime and p ≠ q. Show that has 4 solutions. 6. Find the 4 solutions to note that 493 = 17 × 29. 7. Take an integer n with more than 2 prime factors. Consider: Under what condition is there a solution? Explain thoroughly.