/------------------------------------------------------------\ / \ Q0: Make up your own interesting, elegant, NT problem, then solve it. Aesthetics counts. \ / \____________________________________________________________/ /------------------------------------------------------------\ / \ Show all reasoning for the following problem. Explain computations if any part is obscure. Review the Chinese Remainder Theorem, CRT. Q1: Define 4pos primes P1 := 13, P2 := 17, P3 := 29, and corresponding ronos I1 := 5, I2 := 4, I3 := 12. Let M be the product P1·P2·P3 =NOTE= 6409. Q1a: Use CRT to construct four distinct integers 0 < A < B < C < C < M/2 so that each is an M-rono. [In particular, A² =M= -1.] [Note: Just show the computations for one rono, A, then show me the result, but not the computations, for B,C,D.] Observe that each of ±Ij is a soln to polynomial congruence x²+1 == 0 (mod Pj). Thus CRT will give you 2·2·2=8 solutions [as M-symm-residues] to x²+1 == 0 (mod M). And four of these solutions will be positive. ================ Q1b: Produce four REALLY different intpairs (x, y) such that x² + y² = 6409 . CHOICE: You may either apply SOTS-Euclidean-Algorithm to each of your four M-rono. OR, you can meld together solutions to x² + y² = Pj, for the various primes. Whatever method you choose, explain it carefully for one computation. Question: If you do both SOTS-Euclidean-Algorithm and melding, do you get the same four solutions? ["Same", up to re-ordering and sign change, naturally.] \ / \____________________________________________________________/