Integer Roots

April 19, 2013

One solution first brackets the answer between lo and hi by repeatedly multiplying hi by 2 until n is between lo and hi, then uses binary search to compute the exact answer:

(define (iroot k n)
  (let loop ((hi 1))
    (if (< (expt hi k) n)
        (loop (* hi 2))
        (let loop ((lo (/ hi 2)) (hi hi))
          (let* ((mid (quotient (+ lo hi) 2))
                 (mid^k (expt mid k)))
            (cond ((<= (- hi lo) 1)
                    (if (= (expt hi k) n) hi lo))
                  ((< mid^k n) (loop mid hi))
                  ((< n mid^k) (loop lo mid))
                  (else mid)))))))

A different solution uses Newton’s method, which works perfectly well on integers:

(define (iroot k n)
  (let ((k-1 (- k 1)))
    (let loop ((u n) (s (+ n 1)))
      (if (<= s u) s
        (loop (quotient (+ (* k-1 u) (quotient n (expt u k-1))) k) u)))))

You can run the program at This function will be added to the Standard Prelude the next time it is updated.

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10 Responses to “Integer Roots”

  1. izidor said

    A quick solution using power of 1/k in Haskell:

    iroot :: (Floating b, Integral c, RealFrac b) => b -> b -> c
    iroot k n = floor . exp $ 1/k * log n

  2. OptimusPrime said

    def iroot(k,n):
    for i in range(0,1000):
    if pow(i,k) == n:
    if pow(i,k) n:

  3. eupraxia said

    from itertools import count

    def iroot(k, n):
    ….return next(x for x in count() if x**kn)

  4. eupraxia said

    Ahem. The blog had edited my code?

    from itertools import count

    def iroot(k, n):
    ..return next(x for x in count() if x**kn)

  5. eupraxia said

    Another try. The software doesn’t like less-than or greater-than signs?

    from itertools import count
    from operator import le, gt

    def iroot(k, n):
    ..return next(x for x in count() if le(x**k, n) and gt((x+1)**k, n))

  6. Jussi Piitulainen said

    There’s a link to instructions on how to post source code in the red bar above.

    The source code is embedded in HTML where < and > have to be written &lt; and &gt; (assuming I get this right :) or otherwise escaped.

  7. Jonathan said

    Go Newton’s method! This is a Common Lisp one, not using any floating arithmetic.

    (defun ipow (n k) "Takes n to the kth power"
        (unless (and (integerp k) (not (minusp k))) (error "Power must be a non-negative integer"))
        (unless (and (integerp n) (not (minusp n))) (error "N must be a non-negative integer"))
        (case k (0 1) (1 n) (2 (* n n))
            (t (if (evenp k)
                    (ipow (* n n) (/ k 2))
                    (* n (ipow n (1- k)))))))
    (defun iroot (n &optional (k 2))
        (unless (and (integerp n) (plusp n)) (error "N must be a positive integer"))
        (unless (and (integerp k) (plusp k)) (error "Power must be a positive integer"))
        (loop with 1-1/k = #M 1-1/k
               and k-1 = #M k-1
               and 1/k = #M 1/k
              repeat 1000
              for p = 0 then x
              and x = (floor n (* k k-1)) then (floor (+ (* 1-1/k x) (/ (* n 1/k) (ipow x k-1))))
            ;do (format t "~A -> ~A~%" p x)
            until (or (= x p) (and (<= (ipow x k) n) (<= n (ipow (1+ x) k))))
            finally (return x)))
    ; (iroot 10000 10); => 3
    ; (iroot 10 2); => 3
    ; (iroot 124 3)
    ; (iroot 125 3)
    ; (iroot 126 3)

    It may have a bug – I don’t know if it always converges.

  8. Jeff said

    def iroot(k, n):
    x = float(n)**(1/float(k))
    return int(x)

  9. Shail Shah said

    #define e 2.718281828
    int x,y;
    scanf(“%d %d”, &x,&y);
    void iroot(int m,int n)
    float z;
    int root;

  10. David said

    Did in SmallTalk; the initial guess for Newton’s method is taken by calculating 2 ^ (floor(log2(n)) / k).

    Integer extend [
        floorRoot: n [ |a x0 x y|
            a := self abs.
            x0 := 1 bitShift: a highBit // n.
            x := a // (x0 raisedTo: n-1) - x0 // n + x0.
            [ y := a // (x raisedTo: n-1) - x // n + x.  x > y] whileTrue:
                [ x := y ].
            ^ self > 0 ifTrue: [x] ifFalse: [n odd ifTrue: [x negated] ifFalse: [x i]].
    GNU Smalltalk ready
    st> 1000 floorRoot: 2
    st> 1000 floorRoot: 3
    st> 1000 floorRoot: 4
    st> -125 floorRoot: 3
    st> -25 floorRoot: 2

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