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Exercise 9.3.11. Let n= ab be an odd composite number where a, b e N. Prove that n can be written as the difference of two perfect squares : n =r² - y = (x – y)(x+y),

Exercise 9.3.11. Let n= ab be an odd composite number where a, b e N. Prove that n can be written as the difference of two perfect squares : n =r² - y = (x – y)(x+y),

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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Here is the hint:  Suppose n = ab. Choose a to be the smaller factor. Write
a = x − y and b = x + y, and solve for x and y. To finish the proof, you
need to prove that x and y must both be integers.

Exercise 9.3.11. Let n = ab be an odd composite number where a, b EN. Prove that n can be written as the difference of two perfect squares : n = 2² – y? = (x – y)(x+y), 290 CHAPTER 9 INTRODUCTION TO CRYPTOGRAPHY where both r and y are greater than 1. Consequently, a positive odd integer can be factored exactly when we can find integers r and y such that n = 2² – y?. (*Hint*)
Transcribed Image Text:Exercise 9.3.11. Let n = ab be an odd composite number where a, b EN. Prove that n can be written as the difference of two perfect squares : n = 2² – y? = (x – y)(x+y), 290 CHAPTER 9 INTRODUCTION TO CRYPTOGRAPHY where both r and y are greater than 1. Consequently, a positive odd integer can be factored exactly when we can find integers r and y such that n = 2² – y?. (*Hint*)
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