10. Show that for any non-square d and any integer M there is a finite (possibly empty) set S = {(xi, y₁)} of solutions to x² − dy² = M such that for any solution (a, b) to x²-dy² = M, there exists (x, y) = S such that a+b√d = ±e" (x₂ + y; √d), where e is the fundamental 1-unit. [Hint: mimic the construction, given in lectures, of a solution to x² - dy² = 1 from infinitely many solutions to r²-dy² = M.]
10. Show that for any non-square d and any integer M there is a finite (possibly empty) set S = {(xi, y₁)} of solutions to x² − dy² = M such that for any solution (a, b) to x²-dy² = M, there exists (x, y) = S such that a+b√d = ±e" (x₂ + y; √d), where e is the fundamental 1-unit. [Hint: mimic the construction, given in lectures, of a solution to x² - dy² = 1 from infinitely many solutions to r²-dy² = M.]
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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![10. Show that for any non-square d and any integer M there is a finite (possibly
empty) set S = {(xi, y₁)} of solutions to x² − dy² = M such that for any solution
(a, b) to x²-dy² = M, there exists (x, y) = S such that a+b√d = ±e" (x₂ + y; √d),
where e is the fundamental 1-unit. [Hint: mimic the construction, given in lectures,
of a solution to x² - dy² = 1 from infinitely many solutions to r²-dy² = M.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff76dba93-23d2-47f1-b92c-ad2c0bf14e16%2Fb0d0d577-137a-4bb0-badc-57b23b0a9094%2Fdbt27q_processed.png&w=3840&q=75)
Transcribed Image Text:10. Show that for any non-square d and any integer M there is a finite (possibly
empty) set S = {(xi, y₁)} of solutions to x² − dy² = M such that for any solution
(a, b) to x²-dy² = M, there exists (x, y) = S such that a+b√d = ±e" (x₂ + y; √d),
where e is the fundamental 1-unit. [Hint: mimic the construction, given in lectures,
of a solution to x² - dy² = 1 from infinitely many solutions to r²-dy² = M.]
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