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Re: 2 questions

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  • Aldrich
    ... Well, its certainly a lot more calculation than simply recording a 1 or 2 as each prime candidate in a long chain appears. PARI can check primality of
    Message 1 of 27 , Apr 22, 2011
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      --- In primenumbers@yahoogroups.com, "Phil" <philiplouisbewig@...> wrote:
      >
      > I discussed the Baillie-Wagstaff primality checker at my blog
      > <http://programmingpraxis.com/2010/01/26/primality-checking-revisited/>
      > . Here's the latest version of source code, in Scheme, which differs
      > slightly from the original entry; it's more than "a few lines" of code,
      > but not too long:
      > (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))))))))
      >
      >

      Well, its certainly a lot more calculation than simply recording
      a '1' or '2' as each prime candidate in a long chain appears.
      PARI can check primality of 19-20 digit integers at a rate of
      about 8000/sec, according to Maximillian, so I would expect
      Praxis to be a bit slower. My primitive pm2 hit 11500/sec when
      I put it into a Lazarus FreePascal compiler on a Pentium4, so I
      feel it has potential. I would have to seriously upgrade my
      programming skills to see how far the idea can really go
      however. Somebody please shoot it down - my brain hurts!
      Its pretty easy to check out even if Pascal is not your bag.
      Just download Lazarus and put the program from my Primemine2
      posting in the source editor starting at the line 'type ='.
      Piece of cake.

      Aldrich
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