; 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)