I found the twin-prime-pair $$\large 10^{1000}+9705092\pm 1$$ with PARI/GP.
Is this the smallest twin-prime above $10^{1000}$ ?
A general question to the search of twin primes : The prime number theorem states that the probability, that the number $n$ is prime, is roughly $log(n)$. So, to find a prime above a random prime $n$, you need to check about $log(n)$ numbers to find the next prime.
- But what is the probability that a number $n+2$ is prime, if the number $n$ is prime ? The probabilities are not independent. In particular, if $n>3$ is prime, $n+2$ is odd and the probability that it is divisible by $3$, is $\frac{1}{2}$ and so on. But I do not know how to derive the desired probability from that.
Here is a page on twin primes that gives a possible estimate for the probability that $x$ is the lower of a twin prime pair:
$$\prod_{p\text{ prime}}\frac{p(p-2)}{(p-1)^2}\frac1{(\log x)^2}\\ \approx\frac{0.66016}{(\log x)^2}$$
https://primes.utm.edu/top20/page.php?id=1
The formula was conjectured by Hardy and Littlewood.