## Big Numbers: Examples

### July 5, 2011

All of these exercises are easy, just a matter of substituting the “big” functions for the normal arithmetic operators. Here’s factorial:

`(define (big-factorial n)`

(let loop ((n (- n 1)) (k '(1 1)) (f '(1 1)))

(display (big->string k)) (display " ")

(display (big->string f)) (newline)

(when (positive? n)

(let ((k (big-plus k '(1 1))))

(loop (- n 1) k (big-times f k))))))

`> (big-factorial 50)`

```
```

`1 1`

2 2

3 6

4 24

5 120

6 720

7 5040

8 40320

9 362880

10 3628800

11 39916800

12 479001600

13 6227020800

14 87178291200

15 1307674368000

16 20922789888000

17 355687428096000

18 6402373705728000

19 121645100408832000

20 2432902008176640000

21 51090942171709440000

22 1124000727777607680000

23 25852016738884976640000

24 620448401733239439360000

25 15511210043330985984000000

26 403291461126605635584000000

27 10888869450418352160768000000

28 304888344611713860501504000000

29 8841761993739701954543616000000

30 265252859812191058636308480000000

31 8222838654177922817725562880000000

32 263130836933693530167218012160000000

33 8683317618811886495518194401280000000

34 295232799039604140847618609643520000000

35 10333147966386144929666651337523200000000

36 371993326789901217467999448150835200000000

37 13763753091226345046315979581580902400000000

38 523022617466601111760007224100074291200000000

39 20397882081197443358640281739902897356800000000

40 815915283247897734345611269596115894272000000000

41 33452526613163807108170062053440751665152000000000

42 1405006117752879898543142606244511569936384000000000

43 60415263063373835637355132068513997507264512000000000

44 2658271574788448768043625811014615890319638528000000000

45 119622220865480194561963161495657715064383733760000000000

46 5502622159812088949850305428800254892961651752960000000000

47 258623241511168180642964355153611979969197632389120000000000

48 12413915592536072670862289047373375038521486354677760000000000

49 608281864034267560872252163321295376887552831379210240000000000

50 30414093201713378043612608166064768844377641568960512000000000000

Factorization by trial division first removes factors of 2, then loops through odd numbers starting from 3:

`(define (big-factors n)`

(if (big-even? n)

(cons '(1 2) (factors (big-quotient n '(1 2))))

(let loop ((n n) (f '(1 3)) (fs '()))

(cond ((big-lt? n (big-times f f))

(reverse (cons n fs)))

((big-zero? (big-modulo n f))

(loop (big-quotient n f) f (cons f fs)))

(else (loop n (big-plus f '(1 2)) fs))))))

`(map big->integer (big-factors (integer->big 11111111111111111111)))`

(11 41 101 271 3541 9091 27961)

We’re limiting discussion because these algorithms are all familiar from previous exercises. Here’s the primality checker:

`(define (big-prime? n)`

(define (witness? a n)

(let loop ((r '(0)) (s (big-minus n '(1 1))))

(if (big-even? s)

(loop (big-plus r '(1 1)) (big-quotient s '(1 2)))

(if (big-eq? (big-expm a s n) '(1 1)) #t

(let loop ((j '(0)) (s s))

(cond ((big-eq? j r) #f)

((big-eq? (big-expm a s n) (big-minus n '(1 1))) #t)

(else (loop (big-plus j '(1 1)) (big-times s '(1 2))))))))))

(cond ((big-lt? n '(1 2)) #f)

((big-even? n) (big-eq? n '(1 2)))

(else (let loop ((k 20))

(cond ((zero? k) #t)

((not (witness? (big-rand '(1 1) n) n)) #f)

(else (loop (- k 1))))))))

`> (big-prime? (integer->big (- (expt 2 89) 1)))`

#t

Even though the algorithms are the same, they look quite different because big-numbers are called with function names instead of the normal operators. Here’s the Pollard rho algorithm:

`(define (big-rho n)`

(define (factor n c)

(define (f x) (big-modulo (big-plus (big-times x x) c) n))

(let loop ((x '(1 2)) (y '(1 2)) (d '(1 1)))

(cond ((big-eq? d '(1 1))

(let ((x (f x)) (y (f (f y))))

(loop x y (big-gcd (big-minus x y) n))))

((big-eq? d n) (factor n (big-plus c '(1 1))))

(else d))))

(sort big-lt?

(let fact ((n n) (fs '()))

(cond ((big-eq? n '(1 1)) fs)

((big-even? n) (fact (big-quotient n '(1 2)) (cons '(1 2) fs)))

((big-prime? n) (cons n fs))

(else (let ((f (factor n '(1 1))))

(append fs (fact f '()) (fact (big-quotient n f) '()))))))))

`> (map big->integer (big-rho (integer->big 13290059)))`

(3119 4261)

We used `sort`

from the Standard Prelude. You can run the program at http://programmingpraxis.codepad.org/abIROIpC.

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