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Lets start by defining palindromes and then looking at some problems;

We can define the set $P_d$ as set of all palindromes with $d$ digits.

If $p\in P_d$, then $p=(x,y)$, where $x$ is the numerical value of the palindrome and $y$ is the number base in which we can write $x$ to be palindromic. Then $x,y\in\mathbb N, y\ne1$ .

$$P_d=\{ (x,y) : x=\sum_{n=1}^{d}a_ny^{d-n}, a_n\in\{0\dots y-1\},a_1\ne0,a_n=a_{d-n+1},y\in\mathbb N\setminus\{1\} \}$$

Is there anything out there that one can use to inspect the nature of these sets?
(see question examples below)



Some problem examples

Now, when trying to find palindromes in consecutive number bases, the following questions remain unsolved (and all progress that I'm aware of is based on or comes from computed data):


Q1: Is there a number nontrivially palindromic in $b\ge4$ consecutive number bases?

$(*)$ We can only observe odd $d$ as even palindromes in base $b$ are divisible by $b+1$ .

That is, is there a $P_d$ in which we have pairs $(x,y),(x,y+1),(x,y+2),(x,y+3),\dots$ ?

It would seem that there isn't as $P_3$ seems to be the only set containing infinitely many cases of:
Both pairs $(x,y),(x,y+3)$ along with either of $(x,y+1),(x,y+2)$ for some $x,y$, so far.

And $P_3$ can't have all four pairs (see the question link) - but this is still not fully proven.


Q2: Find all numbers palindromic in $3$ consecutive number bases?

($d=1$ is trivial and thus not being observed as one digit numbers are always palindromes)

I believe I've found all solutions (infinitely many for each $d$ digit case so far) for $d=3,5,7$ digits, but I can't find a single solution for $d\ge9$ digit examples so far.

These are shown in the MO version of the previous question.


Q3: Find all numbers palindromic in $2$ consecutive number bases?

I've found all solutions for $d=3$ and I believe similar expressions can be used for $d=5$, but I have no idea how to fully define the patterns showing in $d\ge7$.

(But this solutions for $d=3$ are still based on computed data, not reached algebraically)

I've tried to observe the gaps between solutions when they are split in number bases $y$ and sorted according to the size of $x$, and noticed a "branching tree like pattern" emerging in $d=7$. A similar pattern emerges in $d=9$, but has much more "branching" spots that occur more often, and when observing the exact gaps, I've managed to extract few patterns but only for the first few out of hundreds more that start appearing as $y$ grows.


Q4: Find all numbers palindromic simultaneously in bases $y,y+2$ ?

Again, I've managed to observe patterns for $d=3,5$ in gaps between the solutions, but I don't see how I can fully define $d\ge7$ so far. (There are some repeating gaps in $d=7,9$, but the pattern isn't fully clear).

You can see the gaps here for $d=3,5,7,9,11$ where "."s represent solutions , and the numbers in [] are gaps - the number of palindromes between the solutions.

(download and open the v2.html in browser to show the text without the linebreaks)



Question

All the progress on these is based on computed data and searching for patterns.

Is there a way to reach (prove) any of the progress on any of these or similar problems algebraically? (without previously observing the data and patterns)

For example, how wold you algebraically reach those found polynomial expressions for $3$ consecutive bases in Q2 for $d=3,5,7$? Why is it so hard to show that those are the only ones that exist?

Simpler example, is there a $d\ge9$ digit number which is simultaneously palindromic in $3$ consecutive number bases? I believe there should be, as $d=3,5,7$ have infinitely many examples.

How can this or similar problems be effectively approached?

Or if not algebraically, can we extract patterns from these problems more effectively?

Or are mathematicians uninterested in these kind of problems, so I'm not finding anything on problems of this nature out there?


P.S. motivation behind this comes out since when I don't know how to solve something completely, but I'm able to get some insights into it, it gets more interesting to waste time on.

Additional examples: is there a number base in which there exists a non palindromic number $x$ whose fifth power is palindromic, $x^5$? Or regarding only the third power and base $10$ and number $2201$? Why is it so hard to prove these observations?

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  • $\begingroup$ Well, I guess you'll just have to find and ask the people who find that interesting. Since "palindrome" depends on the number base, and not even most religions claim that there is a god-given number base (with one exception, but that's in a world I invented, so...), most people don't think that's a relevant property. It should be easy to locate the fifty or so people who don't think so. $\endgroup$ – Professor Vector Dec 16 '17 at 19:45
  • $\begingroup$ @ProfessorVector That's one of the reasons why I'm mostly considering only problems that consider all number bases - to not depend on a single number base. Separation into $d$ digit sets is to make things easier to observe and extend patterns to higher digits once patterns for smaller digits are established, but I'm having trouble in doing so. But I figure this is still mostly uninteresting to general community as you mention. $\endgroup$ – Vepir Dec 16 '17 at 19:53
  • $\begingroup$ Yes, I'm afraid so. BTW, I didn't intend offense, I've answered such questions occasionally... but only if it could be extended to other number bases, like here. $\endgroup$ – Professor Vector Dec 16 '17 at 20:11

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