Smallest Titanic Palindromic Prime

Theorem
The smallest titanic prime that is also palindromic is:
 * $10^{1000} + 81 \, 918 \times 10^{498} + 1$

which can be written as:
 * $1 \underbrace {000 \ldots 000}_{497} 81918 \underbrace {000 \ldots 000}_{497} 1$

Proof
It remains to be demonstrated that it is the smallest such palindromic prime with $1000$ digits or more.

By 11 is Only Palindromic Prime with Even Number of Digits, there are no palindromic primes with exactly $1000$ digits.

Hence such a prime must be greater than $10^{1000}$.

We need to check all numbers of the form:
 * $1 \underbrace {000 \ldots 000}_{497} abcba \underbrace {000 \ldots 000}_{497} 1$

with $\sqbrk {abc} < 819$.

Using the Alpertron integer factorization calculator and the argument: x=0;x=x+1;x<820;10^1000+x*10^500+RevDigits(x/10+10^499,10)

it is verified that there are no primes in the range $\sqbrk {abc} < 819$.

Therefore the number above is the smallest titanic palindromic prime.