Can we prove that a base $b$ must necessarily exist so that only a finite number of palindromes in base $b$ are primes?
My friend came up with this, I hope it isn't one of those ubiquitous "innocent" number theory problems which are years beyond any hope for solution.
So far our ideas have not yielded any fruits:
- The number of palindromes under $n$ is roughly $\sqrt{n}$
- The number of primes under $n$ is roughly $\frac{n}{\log n}$
- Any such prime must have odd length (work $\bmod b+1$)