> Can a natural number 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 was asked on Math.SE](https://math.stackexchange.com/questions/2234587/can-a-number-be-palindrome-in-4-consecutive-number-bases) - but was not solved and still isn't.

Examples for exactly $3$ consecutive bases are mentioned in this [OEIS sequence](http://oeis.org/A279093).

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**Update:** *(added patterns from initial post, and some more on them)*

I believe it can be proved that a solution for $4$ or more consecutive bases does not exist (if this is the case), if all patterns for $3$ consecutive bases are found, since at that point, it would be possible to show if the patters will or never will extend to a fourth consecutive base. (*As was shown in the comments and then the same claim in the answer of the Math.SE post for 3 digit pattern*)

> Even length palindromes seem to not form consecutive palindromes for
> three or more bases - still don't know how to show why this is true.
> 
> Following that, then we choose to observe odd digit length palindromes
> of $2d+1$ digits, $d\in\mathbb N$, and we have so far:

$$(d=1)$$

All examples for this case are of form: ($n=2k+3, k\in\mathbb N$)

$$\frac{1}{2}(n^3 + 6n^2 + 14n + 11)$$

And palindromic in bases $n+1, n+2, n+3$; Except number $300 = 606_7 = 454_8 = 363_9$

> How to show that this is the only pattern and only exception for this
> case?  <br> (Verified up to base $b$; No new exceptions
> or patterns exist for [$b\lt1400$](https://pastebin.com/agTvyCA4), so far)

$$(d=2)$$

All examples for this case are of form: ($n=4k+40, k\in\mathbb N$)

$$\frac{1}{4}(3n^5 + 30n^4 + 125n^3 + 270n^2 + 307n + 148)$$

And palindromic in bases $n+1, n+2, n+3$; No Exceptions found.

>  How to show that this is the only pattern and that there are no
> exceptions?  <br> (No new exceptions or patterns exist for any numbers
> in  all bases [$b\lt220$](https://pastebin.com/ntHVNwg7), so far)

$$(d=3)$$

For $7$-digit palindromes, I do not see a clear pattern. [All the examples so far](https://pastebin.com/cbwdXDVz) computed below: <br> (for $b\le78$, using [this python code](https://codereview.stackexchange.com/q/179611/105332)), Read columns: [base, decimal value, digits in base]

    9 3360633 7
    13 19987816 7
    15 43443858 7
    22 532083314 7
    26 1778140759 7
    28 2721194733 7
    28 11325719295 7
    36 47622367425 7
    37 19683596522 7
    40 97638433343 7
    42 224678540182 7
    43 265282702996 7
    48 561091062285 7
    49 133256051308 7
    56 326217315210 7
    61 597702412638 7
    62 657158314249 7
    68 1242101453540 7
    73 2055729074336 7
    74 2226313335987 7
    74 6678940007962 7
    76 8029674745361 7
    78 9608108112996 7

These are palindromic in column given base $b$ and bases $b+1,b+2$.

> Can a pattern be found here, or anything that can relate and generate
> these examples so far?

$$(d=4)$$

No examples have been found so far which isn't surprising as I only checked all palindromes in number bases $b\lt39$, so far, and the first example for $d=2$ shows at base $b=45$ as seen above, for comparison. Checking all the bases since $d=3$ surprisingly has first example at $b=9$ as seen above.

But this took five hours with the linked python code to check; where the base $b=38$ alone took one hour. (*checking every single nine digit plindrome*); [Pastebin link](https://pastebin.com/Ybz6JNSA) for time comparison.

$$(d\ge5)$$

Haven't searched for examples, as $d=4$ is already taking a long time per number base. 


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*You can see in the linked code that I'm checking each palindrome, but  if you look at examples for $d=3$, they are almost always increasing, thus a lower bound would speed up the search.*