TITLE: Sheldon Cooper Primes
QUESTION [52 upvotes]: On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be the sexy prime $73$

Sheldon : 
  "The best number is $73$. 
  Why? 
  $73$ is the $21^{\text{st}}$ prime number. 
  Its mirror, $37$, is the $12^{\text{th}}$ 
  and its mirror, $21$, is the product of multiplying $7$ and $3$
  ... and in binary $73$ is a palindrome, $1001001$, which backwards is $1001001$."
Leonard : "$73$ is the Chuck Norris of numbers!"
Sheldon : "Chuck Norris wishes... all Chuck Norris backwards gets you is Sirron Kcuhc!"' 

My question is basically this: Are there any more Sheldon Cooper primes?
But how do I define a Sheldon Cooper Prime? Sheldon emphasizes three aspects of 73

It is an emirp with added mirror properties (ie, the prime's mirror is also a prime with position number mirrored)
A concatenation of the factors of the position number of the prime yields the prime.
Binary representation of the prime is a palindrome

I think having all three properties exist simultaneously in a number is difficult.
So, a prime satisfying the first property is good enough.
So, I define a Sheldon Cooper Prime as an emirp with added mirror properties.
Good Luck finding them :D
Edit: Please find primes with position numbers $>9$.
$2,3,..$ are far too trivial.

REPLY [5 votes]: The proof of the "Sheldon Conjecture" was published a few months ago in the February edition of the American Mathematical Monthly.
https://math.dartmouth.edu/~carlp/sheldon02132019.pdf