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; baillie-wagstaff pseudoprimality test

(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)))))

(define (square? n)
  (let ((n2 (isqrt n)))
    (= (* n2 n2) n)))

(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 (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 (primes n)
  (let ((sieve (make-vector n #t)))
    (let loop ((p 2) (ps (list)))
      (cond ((= n p) (reverse ps))
            ((vector-ref sieve p)
              (do ((i p (+ i p))) ((<= n i))
                (vector-set! sieve i #f))
              (loop (+ p 1) (cons p ps)))
            (else (loop (+ p 1) ps))))))

(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 (chain n u v u2 v2 d q m)
  (let loop ((u u) (v v) (u2 u2) (v2 v2) (q q) (qkd q) (m m))
    (if (zero? m) (values u v qkd)
      (let* ((u2 (modulo (* u2 v2) n))
             (v2 (modulo (- (* v2 v2) (* 2 q)) n))
             (q (modulo (* q q) n))
             (qkd (modulo (* qkd q) n)))
        (if (odd? m)
            (let* ((t1 (* u2 v)) (t2 (* u v2))
                   (t3 (* v2 v)) (t4 (* u2 u d))
                   (u (+ t1 t2)) (v (+ t3 t4)))
              (loop (modulo (quotient (if (odd? u) (+ u n) u) 2) n)
                    (modulo (quotient (if (odd? v) (+ v n) v) 2) n)
                    u2 v2 q qkd (quotient m 2)))
            (loop u v u2 v2 q qkd (quotient m 2)))))))

(define (standard-lucas-pseudoprime? n)
  (call-with-values
    (lambda () (selfridge n))
    (lambda (d p q)
      (if (zero? p) (= n d)
        (call-with-values
          (lambda () (chain n 0 2 1 p d q (/ (+ n 1) 2)))
          (lambda (u v qkd) (zero? u)))))))

(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)
  (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 () (chain n 1 p 1 p d q (quotient t 2)))
              (lambda (u v qkd)
                (if (or (zero? u) (zero? v)) #t
                  (let loop ((r 1) (v v) (qkd qkd))
                    (if (= r s) #f
                      (let* ((v (modulo (- (* v v) (* 2 qkd)) n))
                             (qkd (modulo (* qkd qkd) n)))
                        (if (zero? v) #t (loop (+ r 1) v qkd))))))))))))))

(define prime?
  (let ((ps (primes 100)))
    (lambda (n)
      (and (integer? n) (< 1 n) (not (square? n))
           (let loop ((ps ps))
             (cond ((null? ps) #t)
                   ((zero? (modulo n (car ps))) (= n (car ps)))
                   (else (loop (cdr ps)))))
           (strong-pseudoprime? n 2)
           (strong-pseudoprime? n 3)
           (strong-lucas-pseudoprime? n)))))

(do ((ns (list 83 89 111 323 5777 3825123056546413051 (- (expt 2 89) 1)) (cdr ns))) ((null? ns))
  (let ((n (car ns)))
    (display n) (display " ")
    (display (strong-pseudoprime? n 2)) (display " ")
    (display (strong-pseudoprime? n 3)) (display " ")
    (display (standard-lucas-pseudoprime? n)) (display " ")
    (display (strong-lucas-pseudoprime? n)) (display " ")
    (display (prime? n)) (newline)))

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