Origins of the conjecture on the existence of infinitely many palindromic primes 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:
https://en.wikipedia.org/wiki/Palindromic_prime
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?
 A: Let $N(x)$ and $P(x)$ respectively be the number of palindromes and the number of palindromic primes $\le x$. Banks, Hart and Sakata proved that 
$$
P(x) \sim O\bigg(\frac{N(x)\ln\ln\ln x}{\ln x}\bigg).
$$
In the same paper, the authors conjecture that based on their results that the set of palindromes should behave as “random” integers, thus one might expect that the asymptotic relation 
$$
P(x) = O\bigg(\frac{N(x)}{\ln x}\bigg)
$$
For more details refer to the paper Banks, Hart, Sakata 2008, Almost all palindromes are composite.
Another simple heuristic argument is given by Anthony Quas in the comments of the Mathoverflow thread: https://mathoverflow.net/questions/79113/why-so-difficult-to-prove-infinitely-many-restricted-primes
In the interval $2^n$ to $2^{n+1}$ there are approximately $2^{n/2}$ palindromes. Prime number theorem says that the density of primes in this interval is about $1/n$ so we can expect to see about $c2^{n/2}/n$ palindromic primes in this range. Thus in general you can conjecture that the number of palindromic primes $\le x \sim \frac{\sqrt x}{\ln x}$.
A: One of your tags being math-history it is pertinent to mention a topic concerning a way to generate palindromic numbers whatever be prime or composed.
It seems, according to Jean-Paul Delahaye in his paper “Déconcertantes conjectures” that the origin of this theme is in “Palindrome in addition” of Charles Trigg (Mathematics Magazine, 1967. Vol. 40, pages 26-28) however Delahaye adds “mais peut-être est-elle plus ancienne?”.
Starting from a number $N$ written in base 10, we reverse the order of its digits and add to N the inverted number. By repeating several times if necessary the operation, typically found a palindromic number.
Examples
$$14+41=55\\1048+8401=9449\\1723+3271=4994$$
You not always quickly get this way a palindrome and for example starting from $N = 89$ you need twenty-four iterations to get the palindrome
$$(1)\space89+98=187\\(2)\space187+781=968\\(3)\space968+869=1837\\..........\\..........\\(24)\space1801200002107+7012000021081=8813200023188$$
And the first number not giving a palindrome in a known finite number of iterations is $196$. The record till now (of Waden Van Landingham) with $7\cdot 10^8$ iterations without a palindrome appears, stopped at an integer having more than $3\cdot 10^8$ digits and all this has given place to the following
Conjecture: The number $196$ to undergo the rollover-add operation (i.e. the exposed way)
never produce a palindrome.
