; pseudoprimes to bases 2 and 3
(define range
(case-lambda
((stop) (range 0 stop (if (negative? stop) -1 1)))
((start stop) (range start stop (if (< start stop) 1 -1)))
((start stop step)
(let ((le? (if (negative? step) >= <=)))
(let loop ((x start) (xs (list)))
(if (le? stop x) (reverse xs)
(loop (+ x step) (cons x xs))))))
(else (error 'range "too many arguments"))))
(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 (prime? n) ; baillie-wagstaff
(define ps (list 2 3 5 7 11 13 17 19 23 29 31
37 41 43 47 53 59 61 67 71 73 79 83 89 97))
(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 (square? n)
(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)))))
(let ((m (modulo n 128)))
(if (positive? (bitwise-and (* m #x8bc40d7d) (* m #xa1e2f5d1) #x14020a)) #f
(let ((large-mod (modulo n 3989930175))) ; (* 63 25 11 17 19 23 31)
(and (let ((m (modulo large-mod 63)))
(zero? (bitwise-and (* m #x3d491df7) (* m #xc824a9f9) #x10f14008)))
(let ((m (modulo large-mod 25)))
(zero? (bitwise-and (* m #x1929fc1b) (* m #x4c9ea3b2) #x51001005)))
(let ((m (* #xd10d829a (modulo large-mod 31))))
(zero? (bitwise-and m (+ m #x672a5354) #x21025115)))
(let ((m (modulo large-mod 23)))
(zero? (bitwise-and (* m #x7bd28629) (* m #xe7180889) #xf8300)))
(let ((m (modulo large-mod 19)))
(zero? (bitwise-and (* m #x1b8bead3) (* m #x4d75a124) #x4280082b)))
(let ((m (modulo large-mod 17)))
(zero? (bitwise-and (* m #x6736f323) (* m #x9b1d499) #xc0000300)))
(let ((m (modulo large-mod 11)))
(zero? (bitwise-and (* m #xabf1a3a7) (* m #x2612bf93) #x45854000)))
(let ((root (isqrt n))) (if (= (* root root) n) root #f)))))))
(define (jacobi a m)
(if (not (integer? a)) (error 'jacobi "must be integer")
(if (not (and (integer? m) (positive? m) (odd? m)))
(error 'jacobi "modulus must be odd positive integer")
(let loop1 ((a (modulo a m)) (m m) (t 1))
(if (zero? a) (if (= m 1) t 0)
(let ((z (if (member (modulo m 8) (list 3 5)) -1 1)))
(let loop2 ((a a) (t t))
(if (even? a) (loop2 (/ a 2) (* t z))
(loop1 (modulo m a) a
(if (and (= (modulo a 4) 3)
(= (modulo m 4) 3))
(- t) t))))))))))
(define (strong-pseudoprime? 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 (selfridge n)
(let loop ((d-abs 5) (sign 1))
(let ((d (* d-abs sign)))
(cond ((< 1 (gcd d n)) (values d 0 0))
((= (jacobi d n) -1) (values d 1 (/ (- 1 d) 4)))
(else (loop (+ d-abs 2) (- sign)))))))
(define (lucas p q m n) ; right-to-left
(define (even e o) (if (even? n) e o))
(define (mod n) (if (zero? m) n (modulo n m)))
(let ((d (- (* p p) (* 4 q))))
(let loop ((un 1) (vn p) (qn q) (n (quotient n 2))
(u (even 0 1)) (v (even 2 p)) (k (even 1 q)))
(if (zero? n) (values u v k)
(let ((u2 (mod (* un vn))) (v2 (mod (- (* vn vn) (* 2 qn))))
(q2 (mod (* qn qn))) (n2 (quotient n 2)))
(if (even? n) (loop u2 v2 q2 n2 u v k)
(let* ((uu (+ (* u v2) (* u2 v)))
(vv (+ (* v v2) (* d u u2)))
(uu (if (and (positive? m) (odd? uu)) (+ uu m) uu))
(vv (if (and (positive? m) (odd? vv)) (+ vv m) vv))
(uu (mod (/ uu 2))) (vv (mod (/ vv 2))))
(loop u2 v2 q2 n2 uu vv (* k q2)))))))))
(define (powers-of-two n)
(let loop ((s 0) (n n))
(if (odd? n) (values s n)
(loop (+ s 1) (/ n 2)))))
(define (strong-lucas-pseudoprime? n)
; assumes odd positive integer not a square
(call-with-values
(lambda () (selfridge n))
(lambda (d p q)
(if (zero? p) (= n d)
(call-with-values
(lambda () (powers-of-two (+ n 1)))
(lambda (s t)
(call-with-values
(lambda () (lucas p q n t))
(lambda (u v k)
(if (or (zero? u) (zero? v)) #t
(let loop ((r 1) (v v) (k k))
(if (= r s) #f
(let* ((v (modulo (- (* v v) (* 2 k)) n))
(k (modulo (* k k) n)))
(if (zero? v) #t (loop (+ r 1) v k))))))))))))))
(if (not (integer? n)) (error 'prime? "must be integer"))
(if (or (< n 2) (square? n)) #f
(let loop ((ps ps))
(if (pair? ps)
(if (zero? (modulo n (car ps))) (= n (car ps)) (loop (cdr ps)))
(and (strong-pseudoprime? n 2)
(strong-pseudoprime? n 3)
(strong-lucas-pseudoprime? n))))))
(define (psp23? n)
(and (= (expm 2 (- n 1) n) 1)
(= (expm 3 (- n 1) n) 1)
(not (prime? n))))
(display (filter psp23? (range 2 10000)))