Misconstrued Monomials There exists an equation, assuming n and x are positive,



that expresses the relationship between two monomials, one being a common misrepresentation of the other. Many people make the simple mistake of equating these (i.e. 3x^2 and (3x)^2).

Challenge

Given a positive integer, i, determine and return the solution n and x with the smallest sum as an array [n, x]. In the case of a tie, any solution set is acceptable.

Test Cases

62658722541234765
[15, 11]

202500
[4, 15]

524288
[8, 4]

33044255768277
[13, 9]
Mathematica, 61 bytes

Thanks to miles for saving 2 bytes, plus a whole bunch of bytes I counted for no reason!

Last@Select[{n,(#/n)^(1/n)}~Table~{n,2Log@#},IntegerQ@*Last]&amp;


Computes a table of pairs {n,x}, where x = (i/n)^(1/n), using all possible values of n; keeps only those for which the corresponding x is an integer; then returns the pair with the largest value of n.

Here, "all possible values of n" ranges from 1 to 2*ln(i). This ignores the solution {n,x} = {i,1}, but that's okay since the solution {n,x} = {1,i} will suffice if it's the best choice. So x never needs to get smaller than 2, which means that n*2^n ≤ i, and all such n are less than 2*ln(i).

One can show using calculus that the pair {n,x} that minimizes their sum in this context is the same as the pair {n,x} with largest n (not counting {i,1}). That's why the initial Last is good enough to find the optimal pair.