Find all nonnegative real numbers R such that:

³√(13+√R) + ³√(13-√R) is an integer.

Prove that there are no others.

(In reply to Solution by Brian Smith)

Great solution.  I doubt I ever would have found it. 

I jotted this problem down to work on away from my computers and left out non-negative.  I started searching for these negatives on my calculator and got stuck down the rabbit hole of complex conjugates not realizing the solutions were all going to be rational.

If we don't restrict R, there are an infinite number of solutions, but the rest are negative:
I=5 R=-14812/125
I=6 R=-734174/729
I=7 R=-30289904/21^3
etc.
(I was logging in to share my approximate solutions but now I don't need to.)

Posted by Jer on 2016-07-09 13:50:28