A palindromic prime with respect to a base $b \geq 2$ is a prime number such that, when you reverse its sequence of digits in base $b$, you get the same prime. For example in base $10$, the prime $16661$ is palindromic. See the wiki here for more:
It is a well-known conjecture that there are infinitely many of these in base $10$ (but likely in every base). I tried online to find info on the origins of this conjecture but with no success. I couldn't even find in what century it was first talked about. Perhaps it's a folklore problem? Nevertheless there should at least be some first papers which mention it. Would you know anything about the origins of the conjecture or at least the century in which it first appears in a paper or correspondence between mathematicians? Thanks a lot.
Update: Several comments and answers point out heuristics for the conjecture and a recent paper. Would you also know of an old citable paper or book where the conjecture appears or at least something closely related is said about these primes?