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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.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 67E
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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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