Quote:
Originally Posted by bbb120
please tell me the least twin primes which great than 10^1000

This may already be known, but a cursory search failed to turn it up. I did find a lot of larger pairs of twin primes, though, so a least pair greater than 10^1000 certainly exists.
Judging from, e.g.
Twin Prime Statistics, they could be found as follows:
1) Sieve a "suitably" long interval (10^1000, 10^1000 + N) to eliminate all numbers with "small" factors.
2) Subject any remaining pairs differing by 2, starting with the smallest, to a pseudoprime test.
3) If you're really ambitious, subject any twin pseudoprimes you find to a rigorous primality test until you get a pair of proven primes. With numbers of this size, this might take a while.
Based on asymptotic estimates and my own sloppy estimates, you'd probably have to take N on the order of 10^7 or 10^8 to stand a decent chance of success.
I have no idea what a good sieving limit for "small" factors would be.