Clyde Greeno says: >So, are you asking, in essence, for an upper bound to whatever cardinal numbers mankind (or Sam) can "discover?"
Sam? Don't follow.
But sure, exactly. Its a different way of thinking than mathematicians usually like to engage in, so that's why its interesting. You have to make guesses about some fundamental things .
There are no known efficient formulas for generating primes, which is why this is a news article in the first place. Also, from Wikipedia we see:
"The last 15 record primes were Mersenne primes. Before that was a single non-Mersenne (improving the record by merely 37 digits in 1989), and 17 more Mersenne primes going back to 1952."
Is this pattern going to change? Perhaps, but also perhaps not. Will there be a new technique that changes the game? I dunno. It some sense that doesn't really even matter to the fundamental argument, although it would to the estimate.
But, make all the most conservative hypothesis in that regard, then estimate (however!) the curve that graphs the rising and eventually falling computing power of humankind. Does anyone think it will rise literally forever?
I believe that in historical hindsight there will turn out to be "the largest prime ever discovered by man", not that I expect to be there. But the very thought of it seems a bit off the beaten track.