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programmingpraxis - Scheme, pasted on Jun 3:
; williams p+1 factorization algorithm

(define (ilog b n)
  (if (zero? n) -1
    (+ (ilog b (quotient n b)) 1)))

(define (digits n . args)
  (let ((b (if (null? args) 10 (car args))))
    (let loop ((n n) (d '()))
      (if (zero? n) d
          (loop (quotient n b)
                (cons (modulo n b) d))))))

(define prime-bits #f)

(define (load-primes n file-name)
  (with-input-from-file file-name
    (lambda ()
      (let ((k-max (+ (quotient n 30) (if (zero? (modulo n 30)) 0 1))))
        (set! prime-bits (make-vector k-max))
        (do ((k 0 (+ k 1))) ((= k k-max))
          (vector-set! prime-bits k (char->integer (read-char))))))))

(define max-prime 1000000181)
(load-primes 1000000200 "prime.bits")

(define (next-prime n)
  (define (next-bit n)
    (let ((index (quotient n 30))
          (offset (modulo n 30)))
      (case offset
        ((0)                 (values index 1))
        ((1 2 3 4 5 6)       (values index 2))
        ((7 8 9 10)          (values index 4))
        ((11 12)             (values index 8))
        ((13 14 15 16)       (values index 16))
        ((17 18)             (values index 32))
        ((19 20 21 22)       (values index 64))
        ((23 24 25 26 27 28) (values index 128))
        ((29)                (values (+ index 1) 1)))))
  (define (bit-value offset)
    (case offset
      ((1)   1) ((2)   7) ((4)  11) ((8)   13)
      ((16) 17) ((32) 19) ((64) 23) ((128) 29)))
  (define (last-pair xs)
    (if (null? (cdr xs)) xs
      (last-pair (cdr xs))))
  (define (cycle . xs)
    (set-cdr! (last-pair xs) xs) xs)
  (define (get-wheel n)
    (let ((base (* (quotient n 30) 30))
          (offset (modulo n 30)))
      (case offset
        ((0)                 (values (+ base  1) (cycle 6 4 2 4 2 4 6 2)))
        ((1 2 3 4 5 6)       (values (+ base  7) (cycle 4 2 4 2 4 6 2 6)))
        ((7 8 9 10)          (values (+ base 11) (cycle 2 4 2 4 6 2 6 4)))
        ((11 12)             (values (+ base 13) (cycle 4 2 4 6 2 6 4 2)))
        ((13 14 15 16)       (values (+ base 17) (cycle 2 4 6 2 6 4 2 4)))
        ((17 18)             (values (+ base 19) (cycle 4 6 2 6 4 2 4 2)))
        ((19 20 21 22)       (values (+ base 23) (cycle 6 2 6 4 2 4 2 4)))
        ((23 24 25 26 27 28) (values (+ base 29) (cycle 2 6 4 2 4 2 4 6)))
        ((29)                (values (+ base 31) (cycle 6 4 2 4 2 4 6 2))))))
  (cond ((< n 2) 2) ((< n 3) 3) ((< n 5) 5)
        ((< n max-prime)
          (let-values (((index offset) (next-bit n)))
            (let loop ((index index) (offset offset))
              (cond ((= offset 256) (loop (+ index 1) 1))
                    ((zero? (logand (vector-ref prime-bits index) offset))
                      (loop index (* offset 2)))
                    (else (+ (* index 30) (bit-value offset)))))))
        (else (let-values (((k wheel) (get-wheel n)))
                (let loop ((k k) (wheel wheel))
                  (if (prime? k) k (loop (+ k (car wheel)) (cdr wheel))))))))

(define (prime? n)
  (define (expm b e m)
    (define (m* x y) (modulo (* x y) m))
    (cond ((zero? e) 1)
          ((even? e) (expm (m* b b) (/ e 2) m))
          (else (m* b (expm (m* b b) (/ (- e 1) 2) m)))))
  (define (digits n . args)
    (let ((b (if (null? args) 10 (car args))))
      (let loop ((n n) (d '()))
        (if (zero? n) d
            (loop (quotient n b)
                  (cons (modulo n b) d))))))
  (define (isqrt n)
    (let loop ((x n) (y (quotient (+ n 1) 2)))
      (if (<= 0 (- y x) 1) x
        (loop y (quotient (+ y (quotient n y)) 2)))))
  (define (square? n)
    (let ((n2 (isqrt n)))
      (= n (* n2 n2))))
  (define (jacobi a n)
    (if (not (and (integer? a) (integer? n) (positive? n) (odd? n)))
        (error 'jacobi "modulus must be positive odd integer")
        (let jacobi ((a a) (n n))
          (cond ((= a 0) 0)
                ((= a 1) 1)
                ((= a 2) (case (modulo n 8) ((1 7) 1) ((3 5) -1)))
                ((even? a) (* (jacobi 2 n) (jacobi (quotient a 2) n)))
                ((< n a) (jacobi (modulo a n) n))
                ((and (= (modulo a 4) 3) (= (modulo n 4) 3)) (- (jacobi n a)))
                (else (jacobi n a))))))
  (define legendre jacobi)
  (define (inverse x n)
    (let loop ((x (modulo x n)) (a 1))
      (cond ((zero? x) (error 'inverse "division by zero"))
            ((= x 1) a)
            (else (let ((q (- (quotient n x))))
                    (loop (+ n (* q x)) (modulo (* q a) n)))))))
  (define (miller? n a)
    (let loop ((r 0) (s (- n 1)))
      (if (even? s) (loop (+ r 1) (/ s 2))
        (if (= (expm a s n) 1) #t
          (let loop ((r r) (s s))
            (cond ((zero? r) #f)
                  ((= (expm a s n) (- n 1)) #t)
                  (else (loop (- r 1) (* s 2)))))))))
  (define (chain m f g x0 x1)
    (let loop ((ms (digits m 2)) (u x0) (v x1))
      (cond ((null? ms) (values u v))
            ((zero? (car ms)) (loop (cdr ms) (f u) (g u v)))
            (else (loop (cdr ms) (g u v) (f v))))))
  (define (lucas? n)
    (let loop ((a 11) (b 7))
      (let ((d (- (* a a) (* 4 b))))
        (cond ((square? d) (loop (+ a 2) (+ b 1)))
              ((not (= (gcd n (* 2 a b d)) 1))
                (loop (+ a 2) (+ b 2)))
              (else (let* ((x1 (modulo (- (* a a (inverse b n)) 2) n))
                           (m (quotient (- n (legendre d n)) 2))
                           (f (lambda (u) (modulo (- (* u u) 2) n)))
                           (g (lambda (u v) (modulo (- (* u v) x1) n))))
                      (let-values (((xm xm1) (chain m f g 2 x1)))
                        (zero? (modulo (- (* x1 xm) (* 2 xm1)) n)))))))))
  (cond ((or (not (integer? n)) (< n 2))
          (error 'prime? "must be integer greater than one"))
        ((even? n) (= n 2)) ((zero? (modulo n 3)) (= n 3))
        (else (and (miller? n 2) (miller? n 3) (lucas? n)))))

(define (add a b a-b n) (modulo (- (* a b) a-b) n))

(define (double a n) (modulo (- (* a a) 2) n))

(define (mult k v n)
  (let loop ((ks (cdr (digits k 2))) (x v) (y (double v n)))
    (cond ((null? ks) x)
          ((odd? (car ks)) (loop (cdr ks) (add x y v n) (double y n)))
          (else (loop (cdr ks) (double x n) (add x y v n))))))

(define (pplus1-factor n b1 b2 v)
  (let stage1 ((p 2) (v v))
    (let ((a (ilog p b1)))
      (if (< p b1) (stage1 (next-prime p) (mult (expt p a) v n))
        (let ((g (gcd (- v 2) n))) (if (< 1 g n) g
          (let* ((vr v) (c (+ (* (quotient b1 6) 6) 12))
                 (v6 (mult 6 vr n)) (v12 (mult 12 vr n))
                 (z (modulo (* (- v6 vr) (- v12 vr)) n)))
            (let stage2 ((c c) (x v6) (y v12) (z z))
              (if (< c b2)
                  (let ((x+y (add y v6 x n)))
                    (stage2 (+ c 6) y x+y (modulo (* z (- x+y vr)) n)))
                  (let ((g (gcd z n))) (if (< 1 g n) g #f)))))))))))


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