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.
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: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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