Question

I am trying to implement Fermat’s factorization (Algorithm C in The Art of Computer Programming, Vol. 2). Unfortunately in my edition (ISBN 81-7758-335-2), this algorithm is printed incorrectly. what should be the condition on factor-inner loop below? I am running the loop till y <= n [passed in as limit].

(if (< limit y) 0 (factor-inner x (+ y 2) (- r y) limit))

Is there anyway to avoid this condition altogether, as it will double the speed of loop?

(define (factor n) 
  (let ((square-root (inexact->exact (floor (sqrt n))))) 
    (factor-inner (+ (* 2 square-root) 1)
                  1 
                  (- (* square-root square-root) n)
                  n)))

(define (factor-inner x y r limit)
  (if (= r 0)
    (/ (- x y) 2)
    (begin 
      (display x) (display " ") (display y) (display " ") (display r) (newline)
      ;;(sleep-current-thread 1)
      (if (< r 0)
        (factor-inner (+ x 2) y (+  r x) limit)
        (if (< limit y)
          0
          (factor-inner x (+ y 2) (- r y) limit))))))

Answer 1

The (< limit y) check is not necessary because, worst-case, the algorithm will eventually find this solution:

x = N + 2

y = N

It will then return 1.