Check for a prime number using recursive helper function

Question

I am trying to check if a number is prime using recursion. I was required to use a recursive helper function, but I am not sure how I should implement it.

I think I know

Answer 1

(define (isPrimeHelper x k)
  (if (= x k) #t
  (if (= (remainder x k) 0) #f
     (isPrimeHelper x (+ k 1)))))

(define ( isPrime x )
  (cond
    (( = x 0 ) #f)
    (( = x 1 ) #f)
    (( = x 2 ) #t)
     ( else (isPrimeHelper x 2 ) )))

I prefer this version.

Answer 2

Using a helper simply means that you should split your program in smaller parts, and possibly encapsulate loops with extra parameters in separate procedures - and in Scheme loops are frequently implemented via recursive calls. One (naïve) way to implement the is_prime procedure would be:

(define (is_prime n)
  (cond ((<= n 1) #f)
        ((= n 2) #t)
        ((= (modulo n 2) 0) #f)
        (else (check 3 n))))

; recursive helper
(define (check i n)
  (cond ((> (* i i) n) #t)
        ((= (modulo n i) 0) #f)
        (else (check (+ i 2) n))))

There are many ways to implement this procedure, and many possible optimizations; the above should be enough get you started.