Palindromic Cubes

A friend of mine who is well-aware of my obsession with the integers is responsible for this blog post.

We were waiting for our coffees to arrive when he mentioned that it seems, in general, that if a cube number is palindromic then its cube root is also palindromic.

He threw some data my way to get me a bit hot and flustered. The sequence of palindromic cubes proceeds:

1, 8, 343, 1 331, 1 030 301, 1 367 631, 1 003 003 001, 10 662 526 601, 1 000 300 030 001, …

And then if you look at the numbers that generate these cubes, that is, the numbers that you put to the power of three to get the above numbers, you get this sequence:

1, 2, 7, 11, 101, 111, 1 001, 2 201, 10 001, 10 101,…

And these are all palindromic too!

Actually, wait a minute. My friend informs me that if I actually paid attention I would have realised that 2201 is not a palindrome. But all of the others are.

In fact, 2201 is the only known non-palindrome that cubes to become a palindrome. It seems that basically everything else that cubes to become a palindrome will basically also be a palindrome.

According to our dear friends at OEIS, this has been verified up to $10^{15}$ . But a complete proof remains elusive.