Please do Exercise 9.3.13 part D and E show step by step and please explain.Hints:Hint for (a) Prove by contradiction. Hint for (b) Write m = 2k+1.Hint for (d) Use part (c), part (b), and the distributive law.Hint for (e) This is similar to part(b). where both x and y are greater than 1. Consequently, a positive odd integercan be factored exactly when we can find integers x and y such that n =x² - y². (*Hint*)We can use this fact to factor n by trying different pairs of squares inorder to get n as the difference of the two. Of course, we want to do thissystematically. So we want to see what values of r and y we actually needto check:Exercise 9.3.12. In the formula n = x² - y² = (x − y)(x + y), what is thesmallest possible value for r that needs to be tested? (*Hint*)There are other special conditions that r and y must satisfy:Exercise 9.3.13.(a) Assuming that n is an odd number, show that if x is odd then y is even,and if x is even then y is odd. (*Hint*)(b) Show that for any odd number m, then mod (m², 4) = 1. (*Hint*)(c) Let m = x + y. Show that m is odd, and that we can rewrite n =(x−y)(x+y) as: n = m(m - 2y).(d) Show that ifmod (n. 4) = 1, then y must be even. (*Hint*)(e) Show that if mod (n, 4) = 3, then y must be odd. (*Hint*)The Fermat primality testing scheme is better for finding factors thatare nearly equal. The brute force method of Exercise 9.3.14 is much betterwhen one factor is much bigger than the other one. Fermat's test for primalityEven using various tricks to reduce the number of computations, the bruteforce method requires far too many calculations to be useful for RSA encod-ing. A different algorithm for testing primality is Fermat's factorizationalgorithm, which depends on the following fact:Exercise 9.3.11. Let n = ab be an odd composite number where a, b € N.Prove that n can be written as the difference of two perfect squares :n = x² - y² = (x - y) (x + y),

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Answered: Show that if mod (n, 4) = 1, then y…

Math

Advanced Math

Show that if mod (n, 4) = 1, then y must be even. Show that if mod (n, 4) = 3, then y must be odd.

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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