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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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.
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