I have the following solution:
(defn count-swaps [a]
(letfn [(swap [a i j] ;; looked up letfn online
(assoc a i (nth a j) j (nth a i)))]
(loop [a a num-swaps 0 i 0]
(if (< i (count a))
(let [int-loop (loop [a' a j 0 num-swaps' 0]
(if (< j (dec (count a)))
(if (> (nth a j) (nth a (inc j)))
(recur (swap a' j (inc j)) (inc j) (inc num-swaps'))
(recur a' (inc j) num-swaps'))
[a' num-swaps']))]
(recur (nth int-loop 0) (+ num-swaps (nth int-loop 1)) (inc i)))
[num-swaps (nth a 0) (nth a (dec (count a)))]))))
(let [result (count-swaps [4 2 3 1])]
(prn (str "Array is sorted in " (nth result 0) " swaps.") )
(prn (str "First Element: " (nth result 1)) )
(prn (str "Last Element: " (nth result 2)))
)
For this problem:
https://www.hackerrank.com/challenges/ctci-bubble-sort/problem?h_l=interview&playlist_slugs%5B%5D=interview-preparation-kit&playlist_slugs%5B%5D=sorting
However, upon running submitting the problem, none of the tests pass. I don't know why.
after testing this for about an hour or so, I realized where you're mistaken. Namely, using prn instead of print prints out the quote characters alongside the actual text. This was a surprise to me, since I always thought that these two are interchangeable. If you change your prns to printlns, you should be okay.
The final code that I created which passed all of the tests:
;
; Complete the 'countSwaps' function below.
;
; The function accepts INTEGER_ARRAY a as parameter.
;
(defn count-swaps [a]
(letfn [(swap [a i j] ;; looked up letfn online
(assoc a i (nth a j) j (nth a i)))]
(let [result (loop [a a num-swaps 0 i 0]
(if (< i (count a))
(let [int-loop (loop [a' a j 0 num-swaps' 0]
(if (< j (dec (count a)))
(if (> (nth a j) (nth a (inc j)))
(recur (swap a' j (inc j)) (inc j) (inc num-swaps'))
(recur a' (inc j) num-swaps'))
[a' num-swaps']))]
(recur (nth int-loop 0) (+ num-swaps (nth int-loop 1)) (inc i)))
[num-swaps (nth a 0) (nth a (dec (count a)))]))]
(println (str "Array is sorted in " (nth result 0) " swaps.") )
(println (str "First Element: " (nth result 1)))
(println (str "Last Element: " (nth result 2))))))
(def n (Integer/parseInt (clojure.string/trim (read-line))))
(def a (vec (map #(Integer/parseInt %) (clojure.string/split (clojure.string/trimr (read-line)) #" "))))
(count-swaps a)
Let me know if you need any further clearance on this.
I am trying to write a hyperoperation program in MIT/GNU-Scheme however am having some trouble, I have written individual ones up to n=5 working but would like to make one that functions does them all. I will include some of my failed attempts below.
(define hyp (lambda (n x y)
(hypiter n x x y)))
(define hypiter (lambda (lev an cou lim)
(if (= lev 1) an
(hypiter lev (hyp (- lev 1) an cou) cou lim))))
(define hyper (lambda (n x y)
(hyper n x x y)))
(define hyperiter (lambda (lev an cou lim)
(if (= lev 1) (+ an cou)
(hyper (- lev 1) an cou))))
(define h (lambda (n a b)
(cond
((= n 1) (+ a b))
((= b 1) (- n 1))
(else (h (- n 1) (h (- n 1) a a) (- b 1)))))))
(define hyperoperation (lambda (n a b)
(cond
((= n 0) (+ 1 b))
((and (= n 1) (= b 0)) a)
((and (= n 2) (= b 0)) 0)
((and (>= n 3) (= b 0)) 1)
(else (hyperoperation (- b 1) a (hyperoperation n a (- b 1)))))))
According to the definition in wikipedia, there is an error in the last line of your last definition. It should be:
(else (hyperoperation (- n 1) a (hyperoperation n a (- b 1))))))
instead of:
(else (hyperoperation (- b 1) a (hyperoperation n a (- b 1)))))))
So a possible correct recursive definition could be:
(define (hyperoperation n a b)
(cond ((= n 0) (+ b 1))
((= b 0) (cond ((= n 1) a)
((= n 2) 0)
(else 1)))
(else (hyperoperation (- n 1) a (hyperoperation n a (- b 1))))))
I'm still pretty new to proper Lisp and I'm trying to build a simple, yet at least a bit efficient insertion sort - I would like to switch elements in place, but still have an ability to append to my container afterwards. I took my old C++ implementation:
template<typename Iter>
void insertionSort(Iter begin, Iter end){
for (auto i = begin; i != end; ++i){
for (auto j = i; j != begin && *(std::prev(j)) > *(j); j--){
std::iter_swap(j, std::prev(j));
}
}
}
And created the following code (taking into account that aref and rotatef have fair complexity), but it does not seem to take any effect on the input (UPD: now it simply works improperly), what might be wrong with my solution? I'm returning for testing purposes, should I create a macro in order to avoid pass-by-value?
(defparameter testa (make-array 4 :initial-contents '(2 3 1 5)))
(defun insertion-sort (vect)
(loop for i from 0 to (1- (length vect)) do
(loop for j from i downto 0
until (or (= (1- j) -1) (> (aref vect (1- j)) (aref vect j)))
do (rotatef (aref vect i) (aref vect (1- j))))
)
vect
)
(format t "~a~%" (insertion-sort testa))
UPD: updated the code based on the comments from #jkiiski and #RainerJoswig, the output is still wrong.
In your program there are several problems.
First, the sort does not work since the line:
do (rotatef (aref vect i) (aref vect (1- j))))
should be:
do (rotatef (aref vect j) (aref vect (1- j))))
that is, you have written the variable i instead of j
If you make this correction, you will find that the order is decreasing (I assume that you want an increasing order). This depends on the use of until instead of while.
Finally, there is redundant code. A more simple and efficient version is the following:
(defparameter testa (make-array 4 :initial-contents '(2 3 1 5)))
(defun insertion-sort (vect)
(loop for i from 1 below (length vect)
do (loop for j from i above 0
while (> (aref vect (1- j)) (aref vect j))
do (rotatef (aref vect j) (aref vect (1- j)))))
vect)
(format t "~a~%" (insertion-sort testa))
This parallel the pseudo-code in the wikipedia page of Insertion sort.
If you want to parameterize the sorting predicate, as well as add an optional keyword-based “key” parameter to the function, here is a possible solution:
(defun insertion-sort (vect predicate &key (key #'identity))
(loop for i from 1 below (length vect)
do (loop for j from i above 0
while (funcall predicate
(funcall key (aref vect (1- j)))
(funcall key (aref vect j)))
do (rotatef (aref vect j) (aref vect (1- j)))))
vect)
CL-USER> (insertion-sort testa #'>)
#(1 2 3 5)
CL-USER> (insertion-sort testa #'<)
#(5 3 2 1)
CL-USER> (defparameter testa (make-array 4 :initial-contents '((c 3) (d 2) (b 1) (a 4))))
TESTA
CL-USER> (insertion-sort testa #'string> :key #'car)
#((A 4) (B 1) (C 3) (D 2))
I am new in scheme.
How to find "amicable pais"?
(define (SumCD n)
(define s 1 )
(set! m (quotient n 2))
(while (<= i m)
(if (=(modulo n i) 0)
(set! s (+ s i)))
(set! i (+ i 1))
)
)
And in main program I want to check (if (m=SumCD n) and (n=SumCD m)) then m and n is a amicable pair.
How can I do this?
Excessive use of set! indicates an imperative style of programming, which is usually discouraged in Scheme. Here's a Racket-specific implementation of sum-of-divisors that does not use set! at all.
(define (sum-of-divisors n)
(define-values (q r) (integer-sqrt/remainder n))
(for/fold ((sum (if (and (zero? r) (> q 1)) (add1 q) 1)))
((i (in-range 2 q))
#:when (zero? (modulo n i)))
(+ sum i (quotient n i))))
Equivalent version in standard R6RS/R7RS Scheme, if you're not using Racket:
(define (sum-of-divisors n)
(define-values (q r) (exact-integer-sqrt n))
(let loop ((sum (if (and (zero? r) (> q 1)) (+ q 1) 1))
(i 2))
(cond ((>= i q) sum)
((zero? (modulo n i))
(loop (+ sum i (quotient n i)) (+ i 1)))
(else (loop sum (+ i 1))))))
Note that this is not equivalent to the set!-based version you have. What this code actually does is create an inner function, loop, that gets tail-called with new arguments each time.
Now, we can define amicable? and perfect? accordingly:
(define (amicable? n)
(define sum (sum-of-divisors n))
(and (not (= n sum))
(= n (sum-of-divisors sum))))
(define (perfect? n)
(= n (sum-of-divisors n)))
If you really want to test two numbers to see if they are an amicable pair, you can do this:
(define (amicable-pair? a b)
(and (not (= a b))
(= a (sum-of-divisors b))
(= b (sum-of-divisors a))))
Update for OP's new question about how to use this to find amicable pairs between m and n. First, let's define a variant of amicable? that returns a number's amicable "peer":
(define (amicable-peer n)
(define sum (sum-of-divisors n))
(and (not (= n sum))
(= n (sum-of-divisors sum))
sum))
If you're using Racket, use this:
(define (amicable-pairs-between m n)
(for*/list ((i (in-range m (add1 n)))
(peer (in-value (amicable-peer i)))
#:when (and peer (<= m peer n) (< i peer)))
(cons i peer)))
If you're not using Racket, use this:
(define (amicable-pairs-between m n)
(let loop ((result '())
(i n))
(if (< i m)
result
(let ((peer (amicable-peer i)))
(if (and peer (<= m peer n) (< i peer))
(loop (cons (cons i peer) result) (- i 1))
(loop result (- i 1)))))))
The way this works, is that because lists are built from right-to-left, I've decided to count downward from n through to m, keeping only numbers that have an amicable peer, and where the peer is within range. The (< i peer) check is to ensure that the amicable pair only appears once in the results.
Example:
> (amicable-pairs-between 0 10000)
((220 . 284) (1184 . 1210) (2620 . 2924) (5020 . 5564) (6232 . 6368))
More OP updates (wherein he asked what the difference between a recursive version and an accumulative version is). The version of amicable-pairs-between I wrote above is accumulative. A recursive version would look like this:
(define (amicable-pairs-between m n)
(let recur ((i m))
(if (> i n)
'()
(let ((peer (amicable-peer i)))
(if (and peer (<= m peer n) (< i peer))
(cons (cons i peer) (recur (+ i 1)))
(recur (+ i 1)))))))
Note that there is no result accumulator this time. However, it's not tail-recursive any more.
Your program doesn't work: i is never initialized. And it's very poor style; proper Scheme programs seldom use while or set!. Let's go back to the beginning.
A perfect number is equal to the sum of its proper divisors; for instance, the divisors of 28 are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28, so 28 is a perfect number. Two numbers m and n form an amicable pair if the sum of the divisors of m equals n and the sum of the divisors of n equals m; for instance, 220 has divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 which sum to 284, and 284 has divisors 1, 2, 4, 71, 142 which sum to 220, so 220 and 284 form an amicable pair.
A simple way to compute the divisors of a number n is try each integer from 1 to ⌊n/2⌋ and see if it divides n:
(define (divisors n)
(let loop ((i 1) (ds (list)))
(cond ((< n (+ i i)) (reverse ds))
((zero? (modulo n i))
(loop (+ i 1) (cons i ds)))
(else (loop (+ i 1) ds)))))
> (divisors 220)
(1 2 4 5 10 11 20 22 44 55 110)
> (divisors 284)
(1 2 4 71 142)
> (divisors 36)
(1 2 3 4 6 9 12 18)
Note that we are excluding n from the list of divisors of n; that's what we want when computing amicable pairs, but in some cases you might want to add n to the list of divisors of n. Instead of making a list of divisors, we can compute their sum:
(define (sum-div n)
(let loop ((i 1) (s 0))
(cond ((< n (+ i i)) s)
((zero? (modulo n i))
(loop (+ i 1) (+ s i)))
(else (loop (+ i 1) s)))))
> (sum-div 220)
284
> (sum-div 284)
220
> (sum-div 36)
55
Instead of counting up to ⌊n/2⌋, it is faster to note that divisors appear in pairs, so it is only necessary to count up to the square root of n; be careful when n is a perfect square to include exactly one instance of the square root in the sum:
(define (divisors n)
(let loop ((i 2) (ds (list 1)))
(cond ((<= n (* i i))
(sort < (if (= n (* i i)) (cons i ds) ds)))
((zero? (modulo n i))
(loop (+ i 1) (cons i (cons (/ n i) ds))))
(else (loop (+ i 1) ds)))))
(define (sum-div n)
(let loop ((i 2) (s 1))
(cond ((<= n (* i i))
(if (= n (* i i)) (+ i s) s))
((zero? (modulo n i))
(loop (+ i 1) (+ s i (/ n i))))
(else (loop (+ i 1) s)))))
> (divisors 220)
(1 2 4 5 10 11 20 22 44 55 110)
> (divisors 284)
(1 2 4 71 142)
> (divisors 36)
(1 2 3 4 6 9 12 18)
> (sum-div 220)
284
> (sum-div 284)
220
> (sum-div 36)
55
If you know the prime factorization of n, it is easy to find the divisors of n: simply take the products of the members of the powerset of the factor of n, eliminating duplicates.
(define (but-last xs)
(if (null? xs) (error 'but-last "empty list")
(reverse (cdr (reverse xs)))))
(define (unique eql? xs)
(cond ((null? xs) '())
((null? (cdr xs)) xs)
((eql? (car xs) (cadr xs)) (unique eql? (cdr xs)))
(else (cons (car xs) (unique eql? (cdr xs))))))
(define (power-set xs)
(if (null? xs) (list (list))
(let ((rest (power-set (cdr xs))))
(append (map (lambda (x) (cons (car xs) x)) rest) rest))))
(define (divisors n)
(but-last (unique = (sort <
(map (lambda (xs) (apply * xs))
(power-set (factors n)))))))
> (divisors 220)
(1 2 4 5 10 11 20 22 44 55 110)
> (divisors 284)
(1 2 4 71 142)
> (divisors 36)
(1 2 3 4 6 9 12 18)
It is even easier to find the sum of the divisors of n if you know the prime factorization of n by examining the multiplicities of the factors of n:
(define (sum-div n)
(define (div f x) (/ (- (expt f (+ x 1)) 1) (- f 1)))
(let ((fs (factors n)))
(let loop ((f (car fs)) (fs (cdr fs)) (x 1) (s 1))
(cond ((null? fs) (- (* s (div f x)) n))
((= (car fs) f) (loop f (cdr fs) (+ x 1) s))
(else (loop (car fs) (cdr fs) 1 (* s (div f x))))))))
> (sum-div 220)
284
> (sum-div 284)
220
> (sum-div 36)
55
A simple method to find the factors of a number n uses a prime wheel; this is slow if n is a large prime or semi-prime but reasonable otherwise:
(define (factors n)
(define (last-pair xs) (if (null? (cdr xs)) xs (last-pair (cdr xs))))
(define (cycle . xs) (set-cdr! (last-pair xs) xs) xs)
(let ((wheel (cons 1 (cons 2 (cons 2 (cycle 4 2 4 2 4 6 2 6))))))
(let loop ((n (abs n)) (f 2) (wheel wheel) (fs (list)))
(cond ((< n (* f f)) (if (= n 1) fs (reverse (cons n fs))))
((zero? (modulo n f)) (loop (/ n f) f wheel (cons f fs)))
(else (loop n (+ f (car wheel)) (cdr wheel) fs))))))
Given all this, it is easy to determine if a number n is perfect, or if it is part of an amicable pair:
(define (perfect? n)
(= n (sum-div n)))
(define (amicable? n)
(let ((s (sum-div n)))
(and (< 1 s) (= (sum-div s) n))))
> (perfect? 6)
#t
> (perfect? 28)
#t
> (amicable? 220)
#t
> (amicable? 284)
#t
It is also easy to find the perfect numbers and amicable pairs less than some limit:
(define (perfect limit)
(let loop ((n 2) (ps (list)))
(cond ((< limit n) (reverse ps))
((= n (sum-div n))
(loop (+ n 1) (cons n ps)))
(else (loop (+ n 1) ps)))))
(define (amicable limit)
(let loop ((n 2) (as (list)))
(if (< limit n) (reverse as)
(let ((s (sum-div n)))
(if (and (< n s) (= n (sum-div s)))
(loop (+ n 1) (cons (list n s) as))
(loop (+ n 1) as))))))
> (perfect 10000)
(6 28 496 8128)
> (amicable 10000)
((220 284) (1184 1210) (2620 2924) (5020 5564) (6232 6368))
Instead of factoring each number up to a limit, it is much faster to find the sums of the divisors of all numbers up to a limit by sieving: Make a vector from 1 to the limit, each item initialized to 1. Then, for each i from 2 to the limit, add i to each multiple of i:
(define (make-sum-divs n)
(let ((s (make-vector (+ n 1) 0)))
(do ((i 1 (+ i 1))) ((< n i) s)
(do ((j (+ i i) (+ j i))) ((< n j))
(vector-set! s j (+ i (vector-ref s j)))))))
(define max-sum-div 1000)
(define sum-divs (make-sum-divs max-sum-div))
Given the sieve, it is easy to find perfect numbers and amicable pairs:
(define (perfect limit)
(when (< max-sum-div limit)
(set! max-sum-div limit)
(set! sum-divs (make-sum-divs max-sum-div)))
(let loop ((n 2) (ps (list)))
(cond ((< limit n) (reverse ps))
((= n (vector-ref sum-divs n))
(loop (+ n 1) (cons n ps)))
(else (loop (+ n 1) ps)))))
(define (pairs limit)
(when (< max-sum-div limit)
(set! max-sum-div limit)
(set! sum-divs (make-sum-divs max-sum-div)))
(let loop ((n 2) (as (list)))
(if (< limit n) (reverse as)
(let ((s (vector-ref sum-divs n)))
(if (and (< s max-sum-div) (< n s)
(= n (vector-ref sum-divs s)))
(loop (+ n 1) (cons (list n s) as))
(loop (+ n 1) as))))))
> (perfect 1000000)
(6 28 496 8128)
> (pairs 1000000)
((220 284) (1184 1210) (2620 2924) (5020 5564) (6232 6368)
(10744 10856) (12285 14595) (17296 18416) (63020 76084)
(66928 66992) (67095 71145) (69615 87633) (79750 88730)
(100485 124155) (122265 139815) (122368 123152)
(141664 153176) (142310 168730) (171856 176336)
(176272 180848) (185368 203432) (196724 202444)
(280540 365084) (308620 389924) (319550 430402)
(356408 399592) (437456 455344) (469028 486178)
(503056 514736) (522405 525915) (600392 669688)
(609928 686072) (624184 691256) (635624 712216)
(643336 652664) (667964 783556) (726104 796696)
(802725 863835) (879712 901424) (898216 980984))
The sieving method is much faster than either of the other two methods. On my computer, it takes twelve seconds to compute the amicable pairs less than a million using trial division to find the divisors, and about the same amount of time for the factoring method, but only about a second-and-a-half to sieve the divisor sums to a million and another half-a-second to find the amicable pairs, a total of two seconds.
In addition to amicable pairs, there exist amicable chains that cycle back to the start after more than two items. For instance, the numbers 12496, 14288, 15472, 14536, and 14264 form an amicable chain of length 5, since sum-div(12496) = 14288, sum-div(14288) = 15472, sum-div(15472) = 14536, sum-div(14536) = 14264, and sum-div(14264) = 12496. The program to find amicable chains is a variant of the program to find amicable pairs:
(define (chain n limit)
(when (< max-sum-div limit)
(set! max-sum-div limit)
(set! sum-divs (make-sum-divs max-sum-div)))
(let loop ((s (vector-ref sum-divs n)) (cs (list n)))
(cond ((= s n) (reverse cs))
((not (< n s limit)) (list))
((member s cs) (list))
(else (loop (vector-ref sum-divs s) (cons s cs))))))
(define (chains limit)
(when (< max-sum-div limit)
(set! max-sum-div limit)
(set! sum-divs (make-sum-divs max-sum-div)))
(let loop ((n 2) (cs (list)))
(if (< limit n) (reverse cs)
(let ((c (chain n limit)))
(if (null? c) (loop (+ n 1) cs)
(loop (+ n 1) (cons c cs)))))))
> (sort (lambda (a b) (< (length a) (length b))) (chains 1000000))
((6) (28) (496) (8128) (220 284) (1184 1210) (2620 2924)
(5020 5564) (6232 6368) (10744 10856) (12285 14595)
(17296 18416) (63020 76084) (66928 66992) (67095 71145)
(69615 87633) (79750 88730) (100485 124155) (122265 139815)
(122368 123152) (141664 153176) (142310 168730)
(171856 176336) (176272 180848) (185368 203432)
(196724 202444) (280540 365084) (308620 389924)
(319550 430402) (356408 399592) (437456 455344)
(469028 486178) (503056 514736) (522405 525915)
(600392 669688) (609928 686072) (624184 691256)
(635624 712216) (643336 652664) (667964 783556)
(726104 796696) (802725 863835) (879712 901424)
(898216 980984) (12496 14288 15472 14536 14264)
(14316 19116 31704 47616 83328 177792 295488 629072 589786
294896 358336 418904 366556 274924 275444 243760 376736
381028 285778 152990 122410 97946 48976 45946 22976 22744
19916 17716))
The four perfect numbers form amicable chains of length 1, there are 40 amicable pairs, there is an amicable chain of length 5 mentioned above, and notice the spectacular amicable chain of length 28 that starts at 14316.
I just try to find amicable pairs between M and N
(define (find-amicable-pairs M N)
(< M N)
(define i M)
(define a 0)
(do ()
[(= i N)]
(set! a (sum-of-divisors i))
(if (and(= i (sum-of-divisors a)) (< i a))
(and (display i)
(display " and ")
(display a)
(newline))
#f)
(set! i (+ i 1))))
Thanks for your thoughts on this!
I am new to Scheme. I have tried and implemented probabilistic variant of Rabin-Miller algorithm using PLT Scheme. I know it is probabilistic and all, but I am getting the wrong results most of the time. I have implemented the same thing using C, and it worked well (never failed a try). I get the expected output while debugging, but when I run, it almost always returns with an incorrect result. I used the algorithm from Wikipedia.
(define expmod( lambda(b e m)
;(define result 1)
(define r 1)
(let loop()
(if (bitwise-and e 1)
(set! r (remainder (* r b) m)))
(set! e (arithmetic-shift e -1))
(set! b (remainder (* b b) m))
(if (> e 0)
(loop)))r))
(define rab_mil( lambda(n k)
(call/cc (lambda(breakout)
(define s 0)
(define d 0)
(define a 0)
(define n1 (- n 1))
(define x 0)
(let loop((count 0))
(if (=(remainder n1 2) 0)
(begin
(set! count (+ count 1))
(set! s count)
(set! n1 (/ n1 2))
(loop count))
(set! d n1)))
(let loop((count k))
(set! a (random (- n 3)))
(set! a (+ a 2))
(set! x (expmod a d n))
(set! count (- count 1))
(if (or (= x 1) (= x (- n 1)))
(begin
(if (> count 0)(loop count))))
(let innerloop((r 0))
(set! r (+ r 1))
(if (< r (- s 1)) (innerloop r))
(set! x (expmod x 2 n))
(if (= x 1)
(begin
(breakout #f)))
(if (= x (- n 1))
(if (> count 0)(loop count)))
)
(if (= x (- s 1))
(breakout #f))(if (> count 0) (loop count)))#t))))
Also, Am I programming the right way in Scheme? (I am not sure about the breaking out of loop part where I use call/cc. I found it on some site and been using it ever since.)
Thanks in advance.
in general you are programming in a too "imperative" fashion; a more elegant expmod would be
(define (expmod b e m)
(define (emod b e)
(case ((= e 1) (remainder b m))
((= (remainder e 2) 1)
(remainder (* b (emod b (- e 1))) m)
(else (emod (remainder (* b b) m) (/ e 2)))))))
(emod b e))
which avoids the use of set! and just implements recursively the rules
b^1 == b (mod m)
b^k == b b^(k-1) (mod m) [k odd]
b^(2k) == (b^2)^k (mod m)
Similarly the rab_mil thing is programmed in a very non-scheme fashion. Here's an alternative implementation. Note that there is no 'breaking' of the loops and no call/cc; instead the breaking out is implemented as a tail-recursive call which really corresponds to 'goto' in Scheme:
(define (rab_mil n k)
;; calculate the number 2 appears as factor of 'n'
(define (twos-powers n)
(if (= (remainder n 2) 0)
(+ 1 (twos-powers (/ n 2)))
0))
;; factor n to 2^s * d where d is odd:
(let* ((s (twos-powers n 0))
(d (/ n (expt 2 s))))
;; outer loop
(define (loop k)
(define (next) (loop (- k 1)))
(if (= k 0) 'probably-prime
(let* ((a (+ 2 (random (- n 2))))
(x (expmod a d n)))
(if (or (= x 1) (= x (- n 1)))
(next)
(inner x next))))))
;; inner loop
(define (inner x next)
(define (i r x)
(if (= r s) (next)
(let ((x (expmod x 2 n)))
(case ((= x 1) 'composite)
((= x (- n 1)) (next))
(else (i (+ 1 r) x))))
(i 1 x))
;; run the algorithm
(loop k)))