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So we now can take 2 greatest known prime numbers, substract them from some HUGE odd number (bigger than (both these prime numbers) * 2 + 1), and we have a new greatest known prime number?

Or am I missing sth?

EDIT: ah, I get it now, it's sum of SOME 3 prime numbers, but not neccesarily these 2 that I've choosen to make it and another one, it may be 3 completely different primes

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Not every sum is of three primes.

15 = 5 + 5 + 5

which are three primes, but

15 = 2 + 3 + 10

which are not. The theorem says there there is such a sum, not that all sums are of that form.

I don't believe that would work. Because it doesn't specify that it's the sum of ANY three primes. For example, you can't take 21, subtract 7, subtract 5 and say that you have a prime. It has to be two specific primes.

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