We all know the famous Goldbach Conjecture, namely that every even intger $>2$ is the sum of two primes. I was recently playing around with this conjecture and found out that there are MANY ways of writing an even integer as the sum of two primes.
Definition: Let $A(n)$ be the number of ways $n$ can be written as the sum of two primes, where $n$=$0$(mod $2$).
Then we have that Goldbach's conjecture is equivalent to that $A(n)$ is at least $1$.
If it isn't true, please (if possible) provide a counterexample, or some sort of triviality that comes with this statement.
So if you graph this for large enough values, one observes that $A(n)$$\geq$$\sqrt n$. But $\sqrt n$$\geq1$. So would we then have Goldbach's conjecture? In other words, is the following statement true?
Conjecture(?): If one proves $A(n)$$\geq$$\sqrt n$, this would imply $A(n)$$\geq$$1$, which is Goldbach's conjecture.
According to your definition of $A(n)$, Goldbach conjecture would be equivalent to saying that every even number $n >2$ verifies $$A(n) \geq 1.$$ In order to prove Goldbach conjecture, it would be sufficient to prove that for any even number larger than $2$, $A(n) \geq \sqrt{n}$ (this is because $\sqrt{n} \geq 1$ when $n$ is a positive integer).