; shanks' square form factorization algorithm
(define (isqrt n)
(if (not (and (positive? n) (integer? n)))
(error 'isqrt "must be positive integer")
(let loop ((x n))
(let ((y (quotient (+ x (quotient n x)) 2)))
(if (< y x) (loop y) x)))))
(define square?
(let ((q11 (make-vector 11 #f))
(q63 (make-vector 63 #f))
(q64 (make-vector 64 #f))
(q65 (make-vector 65 #f)))
(do ((k 0 (+ k 1))) ((< 5 k))
(vector-set! q11 (modulo (* k k) 11) #t))
(do ((k 0 (+ k 1))) ((< 31 k))
(vector-set! q63 (modulo (* k k) 63) #t))
(do ((k 0 (+ k 1))) ((< 31 k))
(vector-set! q64 (modulo (* k k) 64) #t))
(do ((k 0 (+ k 1))) ((< 32 k))
(vector-set! q65 (modulo (* k k) 65) #t))
(lambda (n)
(if (not (integer? n)) (error 'square? "must be integer")
(if (< n 1) #f
(if (not (vector-ref q64 (modulo n 64))) #f
(let ((r (modulo n 45045)))
(if (not (vector-ref q63 (modulo r 63))) #f
(if (not (vector-ref q65 (modulo r 65))) #f
(if (not (vector-ref q11 (modulo r 11))) #f
(let ((q (isqrt n)))
(if (= (* q q) n) q #f))))))))))))
(define (rho f)
(let* ((a (car f)) (b (cadr f)) (c (caddr f))
(d (- (* b b) (* 4 a c))) (s (isqrt d)) (l (abs c))
(r (if (< l (quotient s 2))
(- (* 2 l (quotient (+ b s) (* 2 l))) b)
(- (* 2 l) b))))
(list c r (quotient (- (* r r) d) (* 4 c)))))
(define (reduce f)
(let* ((a (car f)) (b (cadr f)) (c (caddr f))
(d (- (* b b) (* 4 a c))) (s (isqrt d)))
(if (< (abs (- s (* 2 (abs a)))) b s) f (reduce (rho f)))))
(define (squfof n)
; assumes n is odd, composite, and not a perfect square
(define (enqueue f m q l)
(let* ((x (abs (caddr f))) (g (quotient x (gcd x m))))
(if (and (<= g l) (not (member g q))) (cons g q) q)))
(define (inv-sqrt f c)
(list (- c) (cadr f) (* -1 c (car f))))
(let* ((n4 (= (modulo n 4) 1)) (m (if n4 1 2))
(d (if n4 n (* 4 n))) (s (isqrt d))
(b (if n4 (+ (* 2 (quotient (- s 1) 2)) 1) (* 2 (quotient s 2))))
(delta (quotient (- (* b b) d) 4))
(f (list 1 b delta)) (l (isqrt s)) (bound (* 4 l)))
(let loop ((i 2) (f (rho f)) (q (enqueue f m '() l)))
(cond ((or (< bound i) (= (caddr f) 1)) #f)
((square? (caddr f)) => (lambda (c)
(if (member c q)
(loop (+ i 1) (rho f) (enqueue f m q l))
(let pool ((g (reduce (inv-sqrt f c))))
(let ((new-g (rho g)))
(if (= (cadr g) (cadr new-g))
(let ((c (abs (caddr g))))
(if (odd? c) c (/ c 2)))
(pool new-g)))))))
(else (loop (+ i 1) (rho f) (enqueue f m q l)))))))
(display (squfof 633003781))