# Given (a, b) compute the maximum value of k such that a^{1/k} and b^{1/k} are whole numbers

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Category：Languages

I'm writing a program in C++ that tries to find the minimum value of k > 1 such that the kth root of a and b (which are both given) equals a whole number.

Here's a snippet of my code, which I've commented for clarification.

``int main() {     // Declare the variables a and b.     double a;     double b;     // Read in variables a and b.     while (cin >> a >> b) {          int k = 2;          // We require the kth root of a and b to both be whole numbers.         // "while a^{1/k} and b^{1/k} are not both whole numbers..."         while ((fmod(pow(a, 1.0/k), 1) != 1.0) || (fmod(pow(b, 1.0/k), 1) != 0)) {          k++;          } ``

Pretty much, I read in (a, b), and I start from k = 2 and increment k until the kth roots of a and b are both congruent to 0 mod 1 (meaning that they are divisible by 1 and thus whole numbers).

But, the loop runs infinitely. I've tried researching, and I think it might have to do with precision error; however, I'm not too sure.

Another approach I've tried is changing the loop condition to check whether the floor of a^{1/k} equals a^{1/k} itself. But again, this runs infinitely, likely due to precision error.

Does anyone know how I can fix this issue?

EDIT: for example, when (a, b) = (216, 125), I want to have k = 3 because 216^(1/3) and 125^(1/3) are both integers (namely, 5 and 6).

Floating numbers are not mathematical real numbers. The computation is "approximate". See http://floating-point-gui.de/

You could replace the test `fmod(pow(a, 1.0/k), 1) != 1.0` with something like `fabs(fmod(pow(a, 1.0/k), 1) - 1.0) > 0.0000001` (and play with various such 𝛆 instead of `0.0000001`; see also std::numeric_limits::epsilon but use it carefully, since `pow` might give some error in its computations, and `1.0/k` also inject imprecisions - details are very complex, dive into IEEE754 specifications).

Of course, you could (and probably should) define your `bool almost_equal(double x, double y)` function (and use it instead of `==`, and use its negation instead of `!=`).

As a rule of thumb, never test floating numbers for equality (i.e. `==`), but consider instead some small enough distance between them; that is, replace a test like `x == y` (respectively `x != y`) with something like `fabs(x-y) < EPSILON` (respectively `fabs(x-y) > EPSILON`) where `EPSILON` is a small positive number, hence testing for a small L1 distance (for equality, and a large enough distance for inequality).

And avoid floating point in integer problems.

Actually, predicting or estimating floating point accuracy is very difficult. You might want to consider tools like CADNA. My colleague Franck Védrine is an expert on static program analyzers to estimate numerical errors (see e.g. his TERATEC 2017 presentation on Fluctuat). It is a difficult research topic, see also D.Monniaux's paper the pitfalls of verifying floating-point computations etc.

And floating point errors did in some cases cost human lives (or loss of billions of dollars). Search the web for details. There are some cases where all the digits of a computed number are wrong (because the errors may accumulate, and the final result was obtained by combining thousands of operations)! There is some indirect relationship with chaos theory, because many programs might have some numerical instability.