PMATH 340 Number Theory, Assignment 5 Due: Tues Aug 4 Read Chapter 8 in the Lecture Notes, and work on the Exercises for Chapter 8 in the Practice Problems. Then solve each of the following problems. 1: (a) Let n = 2020. List all of the pairs (x, y) ∈ Z2 such that x2 − y2 = n. (b) Let n = ( 30 7 ) . Find the number of pairs (x, y) ∈ Z2 such that x2 + y2 = n2. (c) Write 650 as a product of irreducible elements in Z[i], then list all of the pairs (x, y) ∈ Z2 with 0 ≤ x ≤ y such that x2 + y2 = 650. 2: (a) Find all solutions (x, y) ∈ Z2 to Pell’s Equation x2 − 29 y2 = 1. (b) Find all solutions (x, y) ∈ Z2 to the Pell-like equation x2 − 21 y2 = 4. 3: The alpha curve in R2 is the set A = { (x, y) ∈ R2 ∣∣ y2 = x3 + x2}. Define f : A \ {(0, 0)} → R \ {1,−1} as follows: given (0, 0) 6= (a, b) ∈ A let f(a, b) = u where u is the (unique) real number such that the point (1, u) lies on the line through (0, 0) and (a, b). Define g : R \ {1,−1} → A \ {(0, 0)} as follows: given u ∈ R with u 6= ±1, let g(u) = (a, b) where (a, b) 6= (0, 0) be the (unique) non-zero point on A which lies on the line through (0, 0) and (1, u). (a) Find a formula for f(a, b) and a formula for g(u) and show that f and g are inverses. (b) Show that the points (x, y) ∈ Q2 with y2 = x3 + x2 are given by (x, y) = (u2 − 1, u3 − u) with u ∈ Q. (c) Show that the points (x, y) ∈ Z2 with y2 = x3 + x2 are given by (x, y) = (u2 − 1, u3 − u) with u ∈ Z. (d) Let p be a prime number. Determine the number of points (x, y) ∈ Zp2 with y2 = x3 + x2.
