If Goldbach's conjecture is false, is it possible that there are only a finite number of failing cases?
I know it is probably unknown, but any reference to something addressing this question would be apreciated.
2026-03-26 06:34:43.1774506883
If Goldbach's conjecture is false, is it possible that there are only a finite number of failing cases?
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You may wish to have a look at section 3.2 of Vaughan's The Hardy-Littlewood Method, where he proved that
Theorem: Let $E(x)$ denote the number of even numbers $\le x$ that cannot be written as a sum of two primes. Then for all $A>0$, there exists a constant $C(A)>0$ such that for large $x$:
$$ E(x)<C(A){x\over\log^Ax} $$
This might not be the best possible bound as the book was published in the 1980s.