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 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 ``` ```; the nth prime ;;;;; small-p and small-pi for base of recursion (define (primes n) (let* ((max-index (quotient (- n 3) 2)) (v (make-vector (+ 1 max-index) #t))) (let loop ((i 0) (ps '(2))) (let ((p (+ i i 3)) (startj (+ (* 2 i i) (* 6 i) 3))) (cond ((>= (* p p) n) (let loop ((j i) (ps ps)) (cond ((> j max-index) (reverse ps)) ((vector-ref v j) (loop (+ j 1) (cons (+ j j 3) ps))) (else (loop (+ j 1) ps))))) ((vector-ref v i) (let loop ((j startj)) (if (<= j max-index) (begin (vector-set! v j #f) (loop (+ j p))))) (loop (+ 1 i) (cons p ps))) (else (loop (+ 1 i) ps))))))) (define max-pi #e32e5) (define ps (list->vector (cons #f (primes max-pi)))) (define max-p (- (vector-length ps) 1)) (define (p n) (if (< max-p n) (nth-prime n) (vector-ref ps n))) (define (pi n) (if (< max-pi n) (prime-pi n) (let loop ((lo 1) (hi (- (vector-length ps) 1))) (let ((mid (quotient (+ lo hi) 2))) (cond ((< hi lo) mid) ((< n (p mid)) (loop lo (- mid 1))) ((< (p mid) n) (loop (+ mid 1) hi)) (else mid)))))) ;;;;; calculation of Legendre's sum phi(x,a) (define (make-hash hash eql? oops size) (let ((table (make-vector size '()))) (lambda (message . args) (if (eq? message 'enlist) (let loop ((k 0) (result '())) (if (= size k) result (loop (+ k 1) (append (vector-ref table k) result)))) (let* ((key (car args)) (index (modulo (hash key) size)) (bucket (vector-ref table index))) (case message ((lookup fetch get ref recall) (let loop ((bucket bucket)) (cond ((null? bucket) oops) ((eql? (caar bucket) key) (cdar bucket)) (else (loop (cdr bucket)))))) ((insert insert! ins ins! set set! store store! install install!) (vector-set! table index (let loop ((bucket bucket)) (cond ((null? bucket) (list (cons key (cadr args)))) ((eql? (caar bucket) key) (cons (cons key (cadr args)) (cdr bucket))) (else (cons (car bucket) (loop (cdr bucket)))))))) ((delete delete! del del! remove remove!) (vector-set! table index (let loop ((bucket bucket)) (cond ((null? bucket) '()) ((eql? (caar bucket) key) (cdr bucket)) (else (cons (car bucket) (loop (cdr bucket)))))))) ((update update!) (vector-set! table index (let loop ((bucket bucket)) (cond ((null? bucket) (list (cons key (caddr args)))) ((eql? (caar bucket) key) (cons (cons key ((cadr args) key (cdar bucket))) (cdr bucket))) (else (cons (car bucket) (loop (cdr bucket)))))))) (else (error 'hash-table "unrecognized message")) )))))) (define phi (let ((memo (make-hash (lambda (x) (+ (* 100000 (car x)) (cadr x))) equal? #f 999983))) (lambda args (let ((x (car args)) (a (cadr args)) (t (memo 'lookup args))) (cond (t t) ; return memoized value ((= a 1) (let ((t (quotient (+ x 1) 2))) (memo 'insert args t) t)) (else (let ((t (- (phi x (- a 1)) (phi (quotient x (p a)) (- a 1))))) (memo 'insert args t) t))))))) ;;;;; the prime-counting function, by Lehmer (define (iroot k n) ; => m such that m^k <= n < (m+1)^k (if (= n 1) 1 (let loop ((hi 1)) (if (< (expt hi k) n) (loop (* hi 2)) (let loop ((lo (/ hi 2)) (hi hi)) (if (= (- hi lo) 1) (if (= (expt hi k) n) hi lo) (let* ((mid (quotient (+ lo hi) 2)) (mid^k (expt mid k))) (cond ((< mid^k n) (loop mid hi)) ((< n mid^k) (loop lo mid)) (else mid))))))))) (define (prime-pi n) (if (< n max-pi) (pi n) (let ((a (pi (iroot 4 n))) (b (pi (iroot 2 n))) (c (pi (iroot 3 n)))) (let i-loop ((i (+ a 1)) (sum (+ (phi n a) (quotient (* (+ b a -2) (- b a -1)) 2)))) (if (< b i) sum (let* ((w (quotient n (p i))) (lim (pi (iroot 2 w))) (sum (- sum (pi w)))) (if (< c i) (i-loop (+ i 1) sum) (let j-loop ((j i) (sum sum)) (if (< lim j) (i-loop (+ i 1) sum) (j-loop (+ j 1) (- sum (pi (quotient w (p j))) (- j) 1))))))))))) ;;;;; approximate pi (define (logint x) (let ((gamma 0.57721566490153286061) (log-x (log x))) (let loop ((k 1) (fact 1) (num log-x) (sum (+ gamma (log log-x) log-x))) (if (< 100 k) sum (let* ((k (+ k 1)) (fact (* fact k)) (num (* num log-x)) (sum (+ sum (/ num fact k)))) (loop k fact num sum)))))) (define (factors n) ; trial division (let loop ((n n) (fs (list))) (if (even? n) (loop (/ n 2) (cons 2 fs)) (if (= n 1) (if (null? fs) (list 1) fs) (let loop ((n n) (f 3) (fs fs)) (cond ((< n (* f f)) (reverse (cons n fs))) ((zero? (modulo n f)) (loop (/ n f) f (cons f fs))) (else (loop n (+ f 2) fs)))))))) (define (mobius-mu n) (if (= n 1) 1 (let loop ((fs (factors n)) (prev 0) (m 1)) (cond ((null? fs) m) ((= (car fs) prev) 0) (else (loop (cdr fs) (car fs) (- m))))))) (define riemann-r (let ((ms (let loop ((n 1) (k 1000) (ms (list))) (if (zero? k) (reverse ms) (let ((m (mobius-mu n))) (if (zero? m) (loop (+ n 1) k ms) (loop (+ n 1) (- k 1) (cons (* m n) ms)))))))) (lambda (x) (let loop ((ms ms) (sum 0)) (if (null? ms) sum (let* ((m (car ms)) (m-abs (abs m)) (m-recip (/ m))) (loop (cdr ms) (+ sum (* m-recip (logint (expt x (/ m-abs)))))))))))) ;;;;; the nth prime (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)))))))) (define (next-prime n) (cond ((< n 2) 2) ((< n 3) 3) (else (let loop ((n (+ (if (even? n) 1 2) n))) (if (prime? n) n (loop (+ n 2))))))) (define (prev-prime n) (cond ((< n 3) #f) ((= n 3) 2) (else (let loop ((n (- n (if (even? n) 1 2)))) (if (prime? n) n (loop (- n 2))))))) (define (nth-prime n) ; (define (approx-pi x) (inexact->exact (round (riemann-r x)))) (define (approx-pi x) (inexact->exact (round (- (logint x) (/ (logint (sqrt x)) 2) (/ (logint (expt x 1/3)) 3))))) (if (< n 5) (vector-ref (vector 2 3 5 7) (- n 1)) (let loop1 ((lo 1) (hi 2)) (if (< (approx-pi hi) n) (loop1 hi (* hi 2)) (let loop2 ((lo lo) (hi hi)) (let* ((mid (quotient (+ lo hi) 2)) (mid-pi (approx-pi mid))) (cond ((< n mid-pi) (loop2 lo mid)) ((< mid-pi n) (loop2 mid hi)) (else (let* ((m (if (prime? mid) mid (prev-prime mid))) (k (prime-pi m))) (cond ((< k n) (let loop3 ((k k) (m m)) (if (= n k) m (loop3 (+ k 1) (next-prime m))))) ((< n k) (let loop3 ((k k) (m m)) (if (= k n) m (loop3 (- k 1) (prev-prime m))))) (else m))))))))))) (display (prime-pi 15485863)) (newline) (display (nth-prime 1000000)) (newline) ```
 ```1 2 ``` ```1000000 15485863 ```