Originally [asked on Math.SE](), but seems like no significant progress took place so far.

*I tend to avoid posting here, but this seems like a question I want to find out an answer to.*


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> Can a natural number $n$ be *nontrivially* [palindromic](https://en.wikipedia.org/wiki/Palindromic_number) in more than $3$  consecutive integer bases? 
> 
> *Nontrivially* means that I'm not counting one-digit palindromes.

The [initial question on Math.SE](https://math.stackexchange.com/questions/2234587/can-a-number-be-palindrome-in-4-consecutive-number-bases) holds all the *progress* so far and references the $3$-digit patterns, but it seems that the exceptions appearing among them prevent a full analysis of the patterns.

I would prefer for the question to be answered **there** with a proof that no three consecutive palindromes will ever expand to a fourth one or more. (Or a counter example)

If there is any other mathematical way to tackle a problem like this one, then I would like to be able to understand the answer if possible, since I'm curious why this seems to be the case.