Those primes are large, not astronomically large as Mersenne primes of course, but there is no such thing as a 'prime table' that you can look it up at this scale, as primes are abundant.
Yes, but clearly it takes less computing power to 'look up' a prime than to use the client software to create it then to check if it is part of a chain.
If you have created a client that can manufacture primes then it is easy to use the client to manufacture them and then store them in a database for analysis.
I am pretty sure that there is such a thing as a 'prime table'. It is a database of known primes.
I am currently investigating this issue among others.