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A positive integer N is said to be a congruent number if it is the area of a right triangle with rational side lengths. For example, 6 is a congruent number because it is the area of the 3 - 4 - 5 triangle and 5 is a congruent number because it is the area of the 3/2 – 20/3 - 41/6 triangle. (a) Let and A = {(X,Y,Z) € Q¹ = }XY = N₁X² + y² = z²¹} E : B = {(x, y) = Q² : y² = x³ - N²x, y ‡ 0} . Show that f(X,Y,Z) = (₁ 242) and g(x, y) = (№2²-2², -23N, N²+2²) -NY 2N² Y Y provide a bijection between the sets A and B.

A positive integer N is said to be a congruent number if it is the area of a right triangle with rational side lengths. For example, 6 is a congruent number because it is the area of the 3 - 4 - 5 triangle and 5 is a congruent number because it is the area of the 3/2 – 20/3 - 41/6 triangle. (a) Let and A = {(X,Y,Z) € Q¹ = }XY = N₁X² + y² = z²¹} E : B = {(x, y) = Q² : y² = x³ - N²x, y ‡ 0} . Show that f(X,Y,Z) = (₁ 242) and g(x, y) = (№2²-2², -23N, N²+2²) -NY 2N² Y Y provide a bijection between the sets A and B.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 39E
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A positive integer N is said to be a congruent number if it is the area of a right triangle with rational side lengths. For example, 6 is a congruent number because it is the area of the 3 - 4 - 5 triangle and 5 is a congruent number because it is the area of the 3/2 20/3 41/6 triangle. (a) Let A = {(X,Y,Z) € Q¹ : }XY = N, X² +Y² = z²} B = {(x, y) ≤ Q² : y² = x³ – N²x, y ‡ 0} . 2N² -2xN Show that f(X,Y,Z) = (NY, 2N2) and g(x, y) = (№²-2², −2ªN¸ №²+2² Y Y y provide a bijection between the sets A and B. and
Transcribed Image Text:A positive integer N is said to be a congruent number if it is the area of a right triangle with rational side lengths. For example, 6 is a congruent number because it is the area of the 3 - 4 - 5 triangle and 5 is a congruent number because it is the area of the 3/2 20/3 41/6 triangle. (a) Let A = {(X,Y,Z) € Q¹ : }XY = N, X² +Y² = z²} B = {(x, y) ≤ Q² : y² = x³ – N²x, y ‡ 0} . 2N² -2xN Show that f(X,Y,Z) = (NY, 2N2) and g(x, y) = (№²-2², −2ªN¸ №²+2² Y Y y provide a bijection between the sets A and B. and
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