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2019-03-09, 14:09   #4
Dr Sardonicus

Feb 2017
Nowhere

1110000000012 Posts

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.

Last fiddled with by Dr Sardonicus on 2019-03-09 at 14:15 Reason: Clarification