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; solovay-strassen primality test (define (square x) (* x x)) (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 rand #f) (define randint #f) (let ((two31 #x80000000) (a (make-vector 56 -1)) (fptr #f)) (define (mod-diff x y) (modulo (- x y) two31)) ; generic version ; (define (mod-diff x y) (logand (- x y) #x7FFFFFFF)) ; fast version (define (flip-cycle) (do ((ii 1 (+ ii 1)) (jj 32 (+ jj 1))) ((< 55 jj)) (vector-set! a ii (mod-diff (vector-ref a ii) (vector-ref a jj)))) (do ((ii 25 (+ ii 1)) (jj 1 (+ jj 1))) ((< 55 ii)) (vector-set! a ii (mod-diff (vector-ref a ii) (vector-ref a jj)))) (set! fptr 54) (vector-ref a 55)) (define (init-rand seed) (let* ((seed (mod-diff seed 0)) (prev seed) (next 1)) (vector-set! a 55 prev) (do ((i 21 (modulo (+ i 21) 55))) ((zero? i)) (vector-set! a i next) (set! next (mod-diff prev next)) (set! seed (+ (quotient seed 2) (if (odd? seed) #x40000000 0))) (set! next (mod-diff next seed)) (set! prev (vector-ref a i))) (flip-cycle) (flip-cycle) (flip-cycle) (flip-cycle) (flip-cycle))) (define (next-rand) (if (negative? (vector-ref a fptr)) (flip-cycle) (let ((next (vector-ref a fptr))) (set! fptr (- fptr 1)) next))) (define (unif-rand m) (let ((t (- two31 (modulo two31 m)))) (let loop ((r (next-rand))) (if (<= t r) (loop (next-rand)) (modulo r m))))) (init-rand 19380110) ; happy birthday donald e knuth (set! rand (lambda seed (cond ((null? seed) (/ (next-rand) two31)) ((eq? (car seed) 'get) (cons fptr (vector->list a))) ((eq? (car seed) 'set) (set! fptr (caadr seed)) (set! a (list->vector (cdadr seed)))) (else (/ (init-rand (modulo (numerator (inexact->exact (car seed))) two31)) two31))))) (set! randint (lambda args (cond ((null? (cdr args)) (if (< (car args) two31) (unif-rand (car args)) (floor (* (next-rand) (car args))))) ((< (car args) (cadr args)) (let ((span (- (cadr args) (car args)))) (+ (car args) (if (< span two31) (unif-rand span) (floor (* (next-rand) span)))))) (else (let ((span (- (car args) (cadr args)))) (- (car args) (if (< span two31) (unif-rand span) (floor (* (next-rand) span)))))))))) (define (jacobi m n) ; also legendre, but not kronecker (cond ((<= n m) (jacobi (modulo m n) n)) ((zero? m) 0) ((= m 1) 1) ((= m 2) (if (even? n) 0 (if (member (modulo n 8) '(1 7)) 1 -1))) ((even? m) (* (jacobi (/ m 2) n) (jacobi 2 n))) ((and (= (modulo m 4) 3) (= (modulo n 4) 3)) (- (jacobi n m))) ; can this be shortened? (else (jacobi n m)))) (define (primes n) (let* ((m (quotient (- n 1) 2)) (sieve (make-vector m #t))) (let loop ((i 0) (p 3) (ps (list 2))) (cond ((= m i) (reverse ps)) ((vector-ref sieve i) (do ((j (/ (- (* p p) 3) 2) (+ j p))) ((<= m j)) (vector-set! sieve j #f)) (loop (+ i 1) (+ p 2) (cons p ps))) (else (loop (+ i 1) (+ p 2) ps)))))) (define (ss-prime? n) (if (not (integer? n)) (error 'ss-prime? "must be integer") (if (< n 2) #f (if (zero? (modulo n 2)) (= n 2) ; n is even (let loop ((k 40) (a (randint 1 n))) (if (zero? k) #t ; probably prime (let ((j (jacobi a n)) (x (expm a (/ (- n 1) 2) n))) (if (or (zero? j) (not (= (modulo j n) x))) #f ; composite (loop (- k 1) (randint 1 n)))))))))) (define (ss-proof? n) (if (not (integer? n)) (error 'ss-proof? "must be integer") (if (< n 2) #f (if (zero? (modulo n 2)) (= n 2) ; n is even (let loop ((as (primes (min (- n 1) (inexact->exact (floor (* 2 (square (log n))))))))) (if (null? as) #t ; prime on the erh (let ((j (jacobi (car as) n)) (x (expm (car as) (/ (- n 1) 2) n))) (if (or (zero? j) (not (= (modulo j n) x))) #f ; composite (loop (cdr as)))))))))) (define-syntax assert (syntax-rules () ((assert expr result) (if (not (equal? expr result)) (for-each display `( #\newline "failed assertion:" #\newline expr #\newline "expected: " ,result #\newline "returned: " ,expr #\newline)))))) (let ((ps (primes 10000))) (do ((n 2 (+ n 1))) ((= n 10000)) (assert (if (member n ps) #t #f) (ss-prime? n)) (assert (if (member n ps) #t #f) (ss-proof? n))))
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