3. Let n = 481 . Do the Miller-Rabin Primality test for the following a: (a) a-8. Show that it re...
Recall that if n is composite, there are at most a 1/4 chance of Miller-Rabin getting a liar. Suppose that we run Miller-Rabin N times on n and each time it thinks n may be prime. Show that the probability that n is prime (approximately) is at least (1 - log(n)/(4^N)). (Hint: it is important here that log(n) is natural log!)
Consider the "candidate prime" n 15841. a. Use Fermat's test ("for a random integer a with 1 < a < n-1: if an-1 1( mod n), then n is composite, else n is probably prime") to check the value of n, taking in turn a = 2, 3, 5, 7. b. Use the Solovay-Strassen test on n with bases a = 2, 3, 5 ("For a random integer a with -) < a < n-1: if ( c. Now use...
Write a computer program that implements the Miller–Rabin algorithm for a user-specified n. The program should allow the user two choices: (1) specify a possible witness to test using the Witness procedure or (2) specify a number s of random witnesses for the Miller–Rabin test to check. Python or C. Sorry to be picky just eaiser to read.
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if 3 od F., then F, is a prime number. (Note: This yields a primality test known as Pepin's Test.)
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number.
Prove that if 3 od F., then F, is a prime number. (Note: This yields a primality test known as Pepin's Test.)
4. (a) [3] Let p be prime and let M, denote the number 2P – 1. The number M, is called a Mersenne number, and if it is prime, it is called a Mersenne prime. There is a test, called the Lucas-Lehmer Test, that gives a necessary and sufficient condition for My to be prime. It is always used to verify that a Mersenne number, suspected of being prime, is indeed a Mersenne prime. Give the statement of this test....
please
complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so
much! (I dont understand your comment what is qs 3.6?)
10.4 Exercise. Show that the algorithm descrihed in Question 3.6 for com puting a (mod n) is a polynomial time algorithm in the number of digits in r In the next scrics of problems you will cxplore the usc of this opcration as a means of testing for primality by starting with a familiar theorem. Theorem (Fermat's Little...
Inverses
3. Let g: N\ { 1 } → N be defined by g(n)-the product of the distinct prime factors of n. a. Find g(2,4, 6, 8, 10, 14)) b. Find g(2, 4, 6, 8, 10, 14)) c. Find g (3)) d. Find g'(1)
3. Let g: N\ { 1 } → N be defined by g(n)-the product of the distinct prime factors of n. a. Find g(2,4, 6, 8, 10, 14)) b. Find g(2, 4, 6, 8, 10, 14))...
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]
8. Let An be the following n x n tridiagonal matrix ab 0 0 0 Cab 00 0 0 Oca 0 0 0. 0 0 0 C a Show that AnalAn- 1l-bc|A,-21 for n 2 3. If a = 1+bc, show that |An 1+bc+ (be)2 ++(bc)" If a 2 cos with 0 <0<T and b c 1 then show that sin (n+1)0 |An = sin 0 nn change
8. Let An be the following n x n tridiagonal matrix ab...
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.