Palindromes in multiple bases

elementary-number-theory decimal-expansion number-systems palindrome

I noticed when listing out palindromes in bases $2$ and $3$ that they seem not to share any palindromes (other than trivial single-digit palindromes). However, when I tried to prove this, I couldn't solve it. I proved it for numbers with an even number of digits in base $3$ by using the fact that a number ending with the digit $0$ is trivially a non-palindrome since leading zeroes don't count as digits, but I can't get it for numbers with an odd number of digits in base $3$.

I also used the fact that $$n\equiv \operatorname{sdig}_b(n) \mod{(b+1)}$$ where $\operatorname{sdig}_b(n)$ is the sum of the digits of $n$ in base $b$.

Can somebody help me prove this?


Solution 1:

You were not able to prove it because it's not true. The OEIS has the first 17 of them as sequence A060792, and it is not known whether any more exist.

Solution 2:

Sadly, $6643_{10}=1100111110011_2=100010001_3$ is a palindrome in bases $2$ and $3$. $1422773$ and $5415589$ also satisfy the property.

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