Question 74
A positive integer $n$ is called doubly reversible of type $\ell$ if there exist two consecutive bases $b$ and $b+1$ such that $n$ is represented by palindromic numbers with $\ell$ digits both in base $b$ and in base $b+1$. For example, 104 is doubly reversible of type 3 because $104_{10}=404_{5}=252_{6}$.
a. Prove that there are no doubly reversible numbers of type 2.
b. Prove that there are infinitely many doubly reversible numbers of type 3.
a. Prove that there are no doubly reversible numbers of type 2.
b. Prove that there are infinitely many doubly reversible numbers of type 3.