Is it possible to construct an arbitrarily long double palindrome?
The double palindrome of length $d$ is a number that is palindromic
(digits are the same when reversed) in two consecutive number bases
$b,b-1$ and has $d\gt 1$ digits in both bases.
Notice that $d$ must be odd. (Even length palindrome in base $b$ is divisible by $b+1$.)
For example, smallest such $d$ length numbers $N$ are:
$$
\begin{array}{llcc}
d & N_{} & N_{b} & N_{b-1} \\
3 & 46 & (1,4,1)_{5} & (2,3,2)_{4} \\
5 & 2293 & (1,4,3,4,1)_{6} & (3,3,1,3,3)_{5} \\
7 & 186621 & (1,4,0,5,0,4,1)_{7} & (3,5,5,5,5,5,3)_{6} \\
9 & 27924649 & (1,5,2,4,1,4,2,5,1)_{8} & (4,5,6,2,3,2,6,5,4)_{7} \\
11 & 1556085529 & (1,3,4,5,7,7,7,5,4,3,1)_{8} & (5,3,3,6,3,3,3,6,3,3,5)_{7}
\end{array}
$$
$$\dots$$
Where $N_b$ stands for number base $b$ representation.
Can we given an arbitrarily large odd $d$, construct such an example? Not necessarily the smallest.
If a construction is not possible, is it possible to have a
non-constructive proof that there exist arbitrarily long double
palindromes?
For example, the following number is a $101$ digit example in number bases $2^{100},2^{100}-1$:
11389275493313395146550195654086875480212234145731621333457701374028277774821274121186469926783503107455762545190548953087972746277002615510348197334563422536978325200285661937560186900957074547554068082502727911310565791405547335060724732113707470568348235577529877640830972500982771607908273897049269199948743133357558899129171595526095424548835696539562402541941975719433140321089322105284423292342890390079652603187050742456213860408145368644790770464116307178226032998988586618940424136245540475050784355875240485281433451060276834218332638393932165203008707194035419270702618571029287812579601921523265433357267147433086934194603149533491309767183140404297760654193824635514373780409273513236609066409655814115873504480016695859332597438995349184138935345329311518673306716195561277801893729959512933999081834483612257653972787850300719280392762476925664658660591935865676106504092843771990798455053144572289465926879848660238840554129637408892668275740988654918664500208238523360411429302322660442324629263685837983291790922905852580315488379578697246636865685154943687657307119964645764231792074703354952892843429147247242575341854166673929009183148029013620039509693002826403446352806308897367164001435010830357381781324567492563737682677932852863861449302117723604251282754369199417086956130386086250554018383792623183489254070735814262747649573875288696676020329121486019334796448294947835513725519213775802399385723069980284364403584079235958069722159900775542477497410968609873477392193126119577904849592080300359176684784985446999145681080782991658907467466272812388989103224984773755050903767298522736370550343965032093005283604035369983437697856001052564882998927925440968051579996174058908430531032383844942218086641153322735698868436889023100943941179461929266276884404712751573931271862837013375482622137967438320352207414572102449928768875364674538369782130207252079580652403427585428426714158838407919917520931159084186491247126021978306309428977838057267458089989192059324625334540178453361150563815452415194771214012690963151049023462937470365410174639417165671169169098495761925964997129692757855110276453683825293816469900688366363665542595611001399702424100153513427148085288952406920565962156464879880387606500753374731675143598406532676463603711230745131611375277036528069799694000409179025588622330937540496488329612388805508117233633052694701641815859674630886375060139622035813116201261468713599560495319754132483733034347504990201455520961778597903897765553458703276959297653931532416792717147421965389813274743401205102119712653419157697182257093836975104016020077311232928824644865884492019118992730353783294077677736829217160116897295006506938648589158119139740497859570466355595233637481562651409130811917086309202404772157419706578610699081034940181844175572714735266695085061024313566678939846144178907828403204463270606610637805786784555542060087712196658611683814223815821199303286564960925262963035771707446370895249357305674148296897358852817848939460321115610826530057710705824101184458195717372478
And it is of size $\approx10^{3040}$ (definitely not the smallest $d=101$ example).
Best Answer
I have not worked out a proof yet , but it seems that $$n:=\frac{b^k-1}{b+1}$$ with even $k\ge 2$ is palindrome in bases $b$ and $b+1$ for sufficient large $b$. For example , $b=10^{99}$ and $k=108$ does the job.