There is a conjecture that there are an infinite number of palindromic prime numbers, but this has not been proved.
However, all the palindromic prime numbers discovered so far have an odd number of digits, except eleven.
Could there be another even-numbered palindromic prime number lurking among the (probably) infinite number of palindromic prime numbers? Or is that impossible, and if so, why?
There is a reasonably well-known proof that if the sum of the odd-numbered digits in a number subtracted from the sum of the even-numbered digits equals zero, the number must be divisible by eleven.
If a palindromic number has an even number of digits then the digits in odd-numbered positions must pair off with numbers in the even-numbered positions. Thus the sum of the odd-numbered digits subtracted from the sum of the even-numbered digits must equal zero. And so the number must be divisible by eleven, and not a prime number.