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Math 346/646 - Homework 1 Assigned: 8/27. Due: 9/3 at the start of class. Notation: Exercise a.b stands for exercise b in Chapter a of Silverman and Tate. Note that the exercises are all together at the end of the chapter. A problem marked with a ∗ is a challenge problem. Problems: 1. Turn in the questionnaire (posted on the class diary) to Jeremy’s office sometime before class on Wednesday. 2. (a) Show that the only integer solutions to w2 = x2 − x + 1 are those with x = 0 and x = 1. (Hint: Write the equation as 4w2 − (4x2 − 4x + 1) = 3.) (b) Find, with proof, all integer solutions to the equation y 2 = x3 − x2 + x. 3. ∗ Prove that there are infinitely many integer solutions to x2 − 5y 2 = 1. Here are the first few solutions: x y 1 0 9 ±4 161 ±72 2889 ±1292 51841 ±23184 930249 ±416020 Based on these examples, try to figure out how to generate the next solution from the previous one, and use induction to prove that there are infinitely many. 1