All the way palindromes

The only nan palindromic number whose cube is a palindrome is 2201. It's cube is 10662526601,which is obviously a palindrome. NOTE:A palindrome is a number which is which means the same when read from both forward and backward.

A non - palindromic number $n$ is that number that is not palindromic (number that remains the same when its digits are reversed) in any number system with a base $x$ in the range 2 ≤ $x$ ≤ $n$ − 2.

To test whether a number n is strictly non-palindromic, it must be verified that n is non-palindromic in all bases up to $n$ − 2.

Until now the only number that has been verified as a non - palindromic number whose cube is a palindromic number is $\boxed{2201}$ .

Its just simple trial and error method in a more complicated way experimented by mathematicians !!!!!

But what can be the solution and reason for this!!!!!!!!!!!!!