Consider the "candidate prime" n 15841. a. Use Fermat's test ("for a random integer a with...
3. Let n = 481 . Do the Miller-Rabin Primality test for the following a: (a) a-8. Show that it returns "probably prime." (b) a-2. Show that it returns "composite." c) a 11. Show that it returns "composite."
3. Let n = 481 . Do the Miller-Rabin Primality test for the following a: (a) a-8. Show that it returns "probably prime." (b) a-2. Show that it returns "composite." c) a 11. Show that it returns "composite."
Recall that an integer >1 is called a prime when its only strictly positive factors are 1 and r. An integer > 1 is called composite when it's not a primec. (a) Show that a composite integer 2 < x < 150 must be a multiple of 2, 3, 5, 7, or 11 (b) Use the Sieve Method and a table with 15 rows and 10 columns to determine all primes between 2 and 150. (c) What's the largest prime...
1 1 point Consider the following algorithm for factoring an integer N provided as input (in binary): For i = 2 to [VN.17 i divides N, then output (i, N/). Which of the following statements is true? This algorithm is correct, but it runs in exponential time. This algorithm is not correct, because it will sometimes fail to find a factorization of Neven if N is composite This algorithm runs in sub-linear time, and always factors N it Nis composite...
A positive integer greater than 1 is said to be prime if it has no divisors other than 1 and itself. A positive integer greater than 1 is composite if it is not prime. Write a program that defines two functions is_prime() and list_primes(). is_prime() will determine if a number is prime. list_primes() will list all the prime numbers below itself other than 1. For example, the first five prime numbers are: 2, 3, 5, 7 and 11." THE PROGRAM...
A Prime Number is an integer which is greater than one, and whose only factors are 1 and itself. Numbers that have more than two factors are composite numbers. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. The number 1 is not a prime number. Write a well-documented, Python program - the main program, that accepts both a lower and upper integer from user entries. Construct a function, isPrime(n), that takes...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
USE PYTHON PLEASE
Write a function called is prime which takes a single integer argument and returns a single Boolean value representing whether the given argument is prime (True) or not (False). After writing the function, test it by using a loop to print out all the prime numbers from 1-100. To check your results, the prime numbers from 1-100 are: 2, 3, 5, 7, 11. 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the prime factorization of 8 is 2 * 2 * 2, so it has one distinct prime divisor. Another: the prime factorization of 12 is 2 * 2 * 3, so it has two distinct prime divisors. A third: 30 = 2 * 3 * 5, which gives it three distinct prime divisors. (n = 24 ⇒ 23768741896345550770650537601358310. From this you conclude that you cannot...
Problem 1: Implement an algorithm to generate prime numbers. You will need to implement the following ingredients (some of them you developed for earlier assignments): 1. A method to generate random binary numbers with n-digits (hint: for the most significant digit, you have no choice, it will be 1; similarly, for the least significant digit there is no choice, it will have to be 1; for all other position, generate 0 or 1 at random) 2. A method to compute...
Problem 3.7. We wish to use the AKS algorithm to test whether 211 is a prime number explanation of AKS. 24 a)211 See the wikipedia reference for a concise ex 1. Use a computer to compute if ( 211a (mod 211) for a suitable choice of a. Be sure to explain why your choice of a is allowed. Does this allow us to conclude -T211 that 211 is prime? 2. Compute our upper bound r < n using the AKS...