Quote:
Originally Posted by LiquidNitrogen
So I guess that means when we "discover" primes of the various forms, they are much larger than the primes that are known to exist in continuity, and therefore they are "real discoveries," in a manner of speaking.

According to the article: "At the time I last updated this page, these projects had found (but not stored) all the prime up to 10^18, but not yet to 10^19."
If you expended 100 times as much effort, you might get up to 10^21. If you devoted the entire world's computer resources to the project, you could probably push it well past 10^30.
You'd never, ever, reach this 100 digit prime:
3664461208681099176204078925954510073897620111029087350504719136242910190767917650858670935504633223
Quote:
If that is true, each prime is probably a unique find.

I don't know what you mean by "unique" in this context. Here's the next one:
3664461208681099176204078925954510073897620111029087350504719136242910190767917650858670935504633509
Both took a fraction of a second to generate on my computer. Neither, in all probability, has ever been "discovered" before.
The primes that are considered "discoveries" are the ones that take significant resources to find
I suggest you read
this primer on primality testing. You'll have a much better understanding of what you see in this forum.