1.2.6 Example: Testing for PrimalityThis section describes two methods for checking the primality of an integer n, one with order of growth , and a probabilistic algorithm with order of growth . The exercises at the end of this section suggest programming projects based on these algorithms. Searching for divisors Since ancient times, mathematicians have been fascinated by problems concerning prime numbers, and many people have worked on the problem of determining ways to test if numbers are prime. One way to test if a number is prime is to find the numbers divisors. The following program finds the smallest integral divisor (greater than 1) of a given number n. It does this in a straightforward way, by testing n for divisibility by successive integers starting with 2. (define (smallest-divisor n)(find-divisor n 2)(define (find-divisor n test-divisor)(cond ( (square test-divisor) n) n)( divides? test-divisor n) test-divisor)(else (find-divisor n (+ test-divisor 1)(define (divides? a b)(= (remainder b a) 0)We can test whether a number is prime as follows: n is prime if and only if n is its own smallest divisor. (define (prime? n)(= n (smallest-divisor n)The end test for find-divisor is based on the fact that if n is not prime it must have a divisor less than or equal to . 44 This means that the algorithm need only test divisors between 1 and . Consequently, the number of steps required to identify n as prime will have order of growth . The Fermat test The primality test is based on a result from number theory known as Fermats Little Theorem. 45 Fermats Little Theorem: If n is a prime number and a is any positive integer less than n, then a raised to the nth power is congruent to a modulo n. (Two numbers are said to be congruent modulo n if they both have the same remainder when divided by n. The remainder of a number a when divided by n is also referred to as the remainder modulo nmodulo n remainder of a modulo n, or simply as a modulo n.) If n is not prime, then, in general, most of the numbers a n will not satisfy the above relation. This leads to the following algorithm for testing primality: Given a number n, pick a random number a n and compute the remainder of an modulo n. If the result is not equal to a, then n is certainly not prime. If it is a, then chances are good that n is prime. Now pick another random number a and test it with the same method. If it also satisfies the equation, then we can be even more confident that n is prime. By trying more and more values of a, we can increase our confidence in the result. This algorithm is known as the Fermat test. To implement the Fermat test, we need a procedure that computes the exponential of a number modulo another number: (define (expmod base exp m)(cond (= exp 0) 1)(even? exp)(remainder (square (expmod base (/ exp 2) m)m)(else(remainder ( base (expmod base (- exp 1) m)m)This is very similar to the fast-expt procedure of section 1.2.4. It uses successive squaring, so that the number of steps grows logarithmically with the exponent. 46 The Fermat test is performed by choosing at random a number a between 1 and n-1 inclusive and checking whether the remainder modulo n of the nth power of a is equal to a. The random number a is chosen using the procedure random, which we assume is included as a primitive in Scheme. Random returns a nonnegative integer less than its integer input. Hence, to obtain a random number between 1 and n - 1, we call random with an input of n - 1 and add 1 to the result: (define (fermat-test n)(define (try-it a)(= (expmod a n n) a)(try-it (+ 1 (random (- n 1)The following procedure runs the test a given number of times, as specified by a parameter. Its value is true if the test succeeds every time, and false otherwise. (define (fast-prime? n times)(cond (= times 0) true)(fermat-test n) (fast-prime? n (- times 1)(else false)Probabilistic methods The Fermat test differs in character from most familiar algorithms, in which one computes an answer that is guaranteed to be correct. Here, the answer obtained is only probably correct. More precisely, if n ever fails the Fermat test, we can be certain that n is not prime. But the fact that n passes the test, while an extremely strong indication, is still not a guarantee that n is prime. What we would like to say is that for any number n, if we perform the test enough times and find that n always passes the test, then the probability of error in our primality test can be made as small as we like. Unfortunately, this assertion is not quite correct. There do exist numbers that fool the Fermat test: numbers n that are not prime and yet have the property that an is congruent to a modulo n for all integers a n. Such numbers are extremely rare, so the Fermat test is quite reliable in practice. 47 There are variations of the Fermat test that cannot be fooled. In these tests, as with the Fermat method, one tests the primality of an intege