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Suppose we have a right-angled triangle and a square of equal area, both of which have side of integer length. Show that this implies the existence of integers p and q with p q and for which p4 - q² is a square. (Hint: you may find Theorem 1.2 in the notes useful here.)

Suppose we have a right-angled triangle and a square of equal area, both of which have side of integer length. Show that this implies the existence of integers p and q with p q and for which p4 - q² is a square. (Hint: you may find Theorem 1.2 in the notes useful here.)

Advanced Engineering Mathematics
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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1. (a) Suppose we have a right-angled triangle and a square of equal area, both of which have side of integer length. Show that this implies the existence of integers p and q with p q and for which p4- q¹ is a square. (Hint: you may find Theorem 1.2 in the notes useful here.) (b) Prove that the equation x4-y4 = z² has no integer solutions. (c) Show that the area of a triangle whose sides have integer length is never the square of an integer.
Transcribed Image Text:1. (a) Suppose we have a right-angled triangle and a square of equal area, both of which have side of integer length. Show that this implies the existence of integers p and q with p q and for which p4- q¹ is a square. (Hint: you may find Theorem 1.2 in the notes useful here.) (b) Prove that the equation x4-y4 = z² has no integer solutions. (c) Show that the area of a triangle whose sides have integer length is never the square of an integer.
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