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) € Q³ : ¹⁄XY = N₁X² + y² = N, Z² = 2²} B = {(x, y) = Q² : y² = x³ − N²x, y ‡ 0} . Show that f(X,Y,Z) = ( (NZ, 242) and g(x, y) = (№²-2², -22 N² +2²) -NY 2N² X+Z¹ X+Z —2ªN¸ Y provide a bijection between the sets A and B. (b) Note that y² = x³ – N²x is an elliptic curve. Use the 3 – 4 – 5 triangle and the map f to find a point P = E(Q). (c) Find the point 2P € E(Q). (d) Use the map g and the point 2P to find another triangle with rational side lengths and area 6.
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) € Q³ : ¹⁄XY = N₁X² + y² = N, Z² = 2²} B = {(x, y) = Q² : y² = x³ − N²x, y ‡ 0} . Show that f(X,Y,Z) = ( (NZ, 242) and g(x, y) = (№²-2², -22 N² +2²) -NY 2N² X+Z¹ X+Z —2ªN¸ Y provide a bijection between the sets A and B. (b) Note that y² = x³ – N²x is an elliptic curve. Use the 3 – 4 – 5 triangle and the map f to find a point P = E(Q). (c) Find the point 2P € E(Q). (d) Use the map g and the point 2P to find another triangle with rational side lengths and area 6.
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
and
1
A = {(X,Y,Z) = Qª³ : ¹ XY = N₁X² + Y² = 2^²}
2
B = {(x, y) ≤ Q² : y² = x³ – N² x, y ‡ 0} .
(N²-2², -2+N N²+2²)
9
Y
Show that f(X,Y,Z) = (XZ, X+2) and g(x, y) =
;) (²
-NY 2N²
X+Z¹
provide a bijection between the sets A and B.
(b) Note that y² = x³ – N²ï is an elliptic curve. Use the 3 – 4 – 5 triangle and
the map f to find a point P = E(Q).
(c) Find the point 2P € E(Q).
(d) Use the map g and the point 2P to find another triangle with rational side
lengths and area 6.
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