Test Prime Using Fermat's Little Theorem

;Fermat's little theorem

 

;If N is a prime number and A is any positive number less than N, then (A^N % N) == (A % N). (A^N stands for A raised to the Nth power.)

 

;This theorem can be used to test the primality of a number. If the above relation is not satisfied for a N, then N is certainly not a prime. If the above relation is satisfied for a N, then the chance is good that N is a prime. The equation is necessary for N to be a prime, but it is not sufficient.

 

;The programs followed are a implementation for testing the primality of a N.

 

;This function call fermat-test several times. ;If failed in a test, then return false. ;Otherwise, return true. (define (prime? n times) (if (= 0 times) true (if (fermat-test n (+ 1 (random (- n 1) ) ) ) (prime? n (- times 1) ) false) ) ) ;Fermat test ;n is the number to be tested ;a is the number less than n and greater than 0 (define (fermat-test n a) (= (exponent-modulo a n n) a) ) ;exponent-modulo (define (exponent-modulo base exp mod) (remainder (exponent base exp) mod) ) ;calculate base^exp (define (exponent base exp) (exponent-helper base exp 1) ) ;tail recursive base^exp helper (define (exponent-helper base exp product) (cond ( (= exp 0) product ) ( (even? exp) (exponent-helper (* base base) (/ exp 2) product) ) ( else (exponent-helper base (- exp 1) (* product base) ) ) ) ) ;determine whether a number is even or odd (define (even? n) (= (remainder n 2) 0) )