; 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)))