Lemma: Any natural number less that $P_{i+1}^2$ is either prime or is divisible by one of $P_{1}, ..., P_{i}$.

If we can show that any even natural number $2n$ which is in the interval $[P_{i}^2, P_{i+1}^2]$ is the sum of two numbers $n-k$ and $n+k$ which are both not divisible by any of $P_{1}, ..., P_{i}$, and where $0\leqslant k< \frac{P_{i}^2}{2}$ then since $n-k$ and $n+k$ must both be prime due to the above lemma, the Goldbach conjecture is true.

Would it be possible to do this using a sieve? Something like the sieve of Eratosthenes, and then show that the number of $k$'s is something like $\frac{1}{2}\pi(\frac{P_{i}^2}{2})$ where $\pi$ is the prime counting function.

So basically, sieve out all numbers of the form $P_{j}l + a_{j}$ and $P_{j}l + (P_{j}-a_{j})$ for integers $l$, and $0\leqslant j\leqslant i$, where $a_{j} = n\mod P_{j}$, and count what's left over.

Any ideas on how to do this?


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