40. This exercise introduces another problem that can be solved by finding rational points on an elliptic curve. Consider a collection of balls arranged in a square pyramid with x square layers, with one ball in the top layer, four in the layer below that, and so on, with x2 in the bottom layer. a) Show that we can rearrange the balls in the pyramid into a single square of side y if and only if there is a positive integer solution (x, y) to y² = x(x + 1)(2x + 1)/6. b) Show that if 1<≤x≤ 10, it is possible to arrange the balls into a square pyramid only when x = 1. c) Show that both (0, 0) and (1, 1) lie on the curve y² = x(x + 1)(2x + 1)/6. Find the sum of (0, 0) and (1, 1) on this curve. d) Find sum of the point you found in part (c) and (1, 1). Show that this sum leads to a positive integer solution.

40. This exercise introduces another problem that can be solved by finding rational points on an elliptic curve. Consider a collection of balls arranged in a square pyramid with x square layers, with one ball in the top layer, four in the layer below that, and so on, with x2 in the bottom layer. a) Show that we can rearrange the balls in the pyramid into a single square of side y if and only if there is a positive integer solution (x, y) to y² = x(x + 1)(2x + 1)/6. b) Show that if 1<≤x≤ 10, it is possible to arrange the balls into a square pyramid only when x = 1. c) Show that both (0, 0) and (1, 1) lie on the curve y² = x(x + 1)(2x + 1)/6. Find the sum of (0, 0) and (1, 1) on this curve. d) Find sum of the point you found in part (c) and (1, 1). Show that this sum leads to a positive integer solution.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

See similar textbooks