; home primes

(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 (undigits ds . args)
  (let ((b (if (null? args) 10 (car args))))
    (let loop ((ds ds) (n 0))
      (if (null? ds) n
          (loop (cdr ds) (+ (* n b) (car ds)))))))

(define (prime? n)
  (letrec (
    (expm (lambda (b e m)
      (let ((times (lambda (x y) (modulo (* x y) m))))
        (cond ((zero? e) 1) ((even? e) (expm (times b b) (/ e 2) m))
        (else (times b (expm (times b b) (/ (- e 1) 2) m)))))))
    (digits (lambda (n)
      (let loop ((n n) (ds '()))
        (if (zero? n) ds (loop (quotient n 2) (cons (modulo n 2) ds))))))
    (isqrt (lambda (n)
      (let loop ((x n) (y (quotient (+ n 1) 2)))
        (if (<= 0 (- y x) 1) x (loop y (quotient (+ y (quotient n y)) 2))))))
    (square? (lambda (n) (let ((n2 (isqrt n))) (= n (* n2 n2)))))
    (jacobi (lambda (a n)
      (let loop ((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) (* (loop 2 n) (loop (/ a 2) n)))
              ((< n a) (loop (modulo a n) n))
              ((and (= (modulo a 4) 3) (= (modulo n 4) 3)) (- (loop n a)))
              (else (loop n a))))))
    (inverse (lambda (x m)
      (let loop ((a 1) (b 0) (g x) (u 0) (v 1) (w m))
        (if (zero? w) (modulo a m)
          (let ((q (quotient g w)))
            (loop u v w (- a (* q u)) (- b (* q v)) (- g (* q w))))))))
    (strong-pseudo-prime? (lambda (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))))))))))
    (chain (lambda (m f g u v)
      (let loop ((ms (digits m)) (u u) (v v))
        (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)))))))
    (lucas-pseudo-prime? (lambda (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 (jacobi 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)))))))))))
    (if (not (integer? n)) (error 'prime? "must be integer")
      (if (< n 2) #f (if (even? n) (= n 2) (if (zero? (modulo n 3)) (= n 3)
        (and (strong-pseudo-prime? n 2)
             (strong-pseudo-prime? n 3)
             (lucas-pseudo-prime? n))))))))

(define (factors n)
  (letrec (
    (last-pair (lambda (xs) (if (pair? (cdr xs)) (last-pair (cdr xs)) xs)))
    (cycle (lambda xs (set-cdr! (last-pair xs) xs) xs))
    (cons* (lambda (x . xs)
      (if (null? xs) x (cons x (apply cons* (car xs) (cdr xs)))))))
    (if (not (integer? n)) (error 'factors "must be integer")
      (if (member n '(-1 0 1)) (cdr (assoc n '((-1 -1) (0 0) (1 1))))
        (if (negative? n) (cons -1 (factors (- n)))
          (let wheel ((n n) (p 2) (ps '()) (ws (cons* 1 2 2 (cycle 4 2 4 2 4 6 2 6))))
            (if (= n 1) (reverse ps)
              (if (< n (* p p)) (reverse (cons n ps))
                (if (zero? (modulo n p)) (wheel (/ n p) p (cons p ps) ws)
                  (if (< p 1000) (wheel n (+ p (car ws)) ps (cdr ws))
                    (let rho ((ps ps) (cs (list n)) (a 1) (m 10000000))
                      (letrec (
                        (f (lambda (y) (modulo (+ (* y y) a) (car cs))))
                        (g (lambda (p x y) (modulo (* p (abs (- x y))) (car cs)))))
                        (if (null? cs) (sort < ps)
                          (let ((n (car cs)))
                            (if (= n 1) (rho ps (cdr cs) a m)
                              (if (prime? n) (rho (cons n ps) (cdr cs) a m)
                                (let loop ((x 2) (y (f 2)) (j 1) (q 2) (p 1))
                                  (if (= j m) (cons (sort < ps) cs)
                                    (if (= j q)
                                        (let ((t (f y)))
                                          (loop y t (+ j 1) (* q 2) (g p y t)))
                                        (if (positive? (modulo j 100))
                                            (loop x (f y) (+ j 1) q (g p x y))
                                            (let ((d (gcd p n)))
                                              (if (= d 1) (loop x (f y) (+ j 1) q 1)
                                                (if (= d n) (rho ps cs (+ a 1) (- m j))
                                                  (rho ps (cons* d (/ n d) (cdr cs))
                                                    a (- m j)))))))))))))))))))))))))

(define (home-prime n)
  (let* ((fs (factors n))
         (hp (undigits (apply append (map digits fs)))))
    (if (prime? hp) hp (home-prime hp))))

(define (home-factors n)
  (let loop ((n n) (hps '()))
    (if (prime? n) (reverse (cons n hps))
      (let* ((fs (factors n))
             (hp (undigits (apply append (map digits fs)))))
        (loop hp (cons fs hps))))))

(display (home-prime 99)) (newline)
(display (home-factors 99)) (newline)

71143
((3 3 11) (7 11 43) 71143)