Project:
 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 ``` ```; primality checking, revisited (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 (digits n . args) (let ((b (if (null? args) 10 (car args)))) (let loop ((n n) (d '())) (if (zero? n) d (loop (quotient n b) (cons (modulo n b) d)))))) (define (isqrt n) (let loop ((x n) (y (quotient (+ n 1) 2))) (if (<= 0 (- y x) 1) x (loop y (quotient (+ y (quotient n y)) 2))))) (define (square? n) (let ((n2 (isqrt n))) (= n (* n2 n2)))) (define (jacobi a n) (if (not (and (integer? a) (integer? n) (positive? n) (odd? n))) (error 'jacobi "modulus must be positive odd integer") (let jacobi ((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) (* (jacobi 2 n) (jacobi (quotient a 2) n))) ((< n a) (jacobi (modulo a n) n)) ((and (= (modulo a 4) 3) (= (modulo n 4) 3)) (- (jacobi n a))) (else (jacobi n a)))))) (define legendre jacobi) (define (inverse x n) (let loop ((x (modulo x n)) (a 1)) (cond ((zero? x) (error 'inverse "division by zero")) ((= x 1) a) (else (let ((q (- (quotient n x)))) (loop (+ n (* q x)) (modulo (* q a) n))))))) (define (miller? 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 (chain m f g x0 x1) (let loop ((ms (digits m 2)) (u x0) (v x1)) (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)))))) (define (lucas? 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 (legendre 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))))))))) (define (prime? n) (cond ((or (not (integer? n)) (< n 2)) (error 'prime? "must be integer greater than one")) ((even? n) (= n 2)) ((zero? (modulo n 3)) (= n 3)) (else (and (miller? n 2) (miller? n 3) (lucas? n))))) (define (mersenne k) (- (expt 2 k) 1)) (do ((k 2 (+ k 1))) ((< 1000 k)) (if (prime? (mersenne k)) (begin (display k) (display #\tab) (display (mersenne k)) (newline)))) ```
 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ``` ```2 3 3 7 5 31 7 127 13 8191 17 131071 19 524287 31 2147483647 61 2305843009213693951 89 618970019642690137449562111 107 162259276829213363391578010288127 127 170141183460469231731687303715884105727 521 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151 607 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127 Timeout```