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.
Ok. Thinking about it there probably isn't a complete database of all known primes but that doesn't mean that there isn't a weakness in your pow scheme.
Is your client software the most efficient way of computing these primes and prime chains and verifying them or can a more efficient mathematical way be produced? I bet you can't show me a mathematical proof of this. Maybe you can??
Whereas the SHA 256 hashing function used in bitcoin has undergone a pretty thorough mathematical analysis by some very clever people over the years.
I'm just being the devil's advocate but these questions need to be asked.