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Edmund Landau had once put forth the following statement about the validity of the Goldbach conjecture.

"the goldbach conjecture is false for at most 0 % of all even integers ; this at most 0 % does not exclude, however the possibility that there are infinitely many exceptions"

I have not understood the essence of the statement, like for an ordinary novice student like me, the two statements appear contradicting, whilst they appear to make perfect sense for a number theorist. Can anyone please help me with the understanding of the essence of this statement?

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Consider this statement: $0\%$ of all natural numbers are powers of $2$. It is true, in this sense: if, for each natural $n$, you compute the proportion $p_n$ of powers of $2$ in $\{1,2,\ldots,n\}$, then $\lim_{n\to\infty}p_n=0$. But that does not mean that no natural number is a power of $2$.

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  • $\begingroup$ Ok, I understood most of it, but what is the importance of the clause "at most" in the statement? $\endgroup$ – saisanjeev Feb 11 '18 at 15:49
  • $\begingroup$ @saisanjeev That I don't know. I suspect that it is not meant to be take literally, but I am not aware of the context. $\endgroup$ – José Carlos Santos Feb 11 '18 at 15:52
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Here's an analogy.

There are infinitely many numbers of the form $10^k$ - a $1$ followed by some number of zeroes. Among the first $10$ numbers there are just $2$. Among the first $100$ numbers there are just three. Among the first $1000$, four. That pattern continues, so the fraction of all numbers of that form might reasonably be said to be $0$, even though there are infinitely many of them.

Whether there are infinitely many counterexamples to Goldbach's conjecture is unknown. We don't even know there's one.

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