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Is it possible to solve this problem using extremely precise approximations in all the asymptotic derivations discussed here? For instance, if $S$ is the set of prime numbers, then $N_S(z) \sim z/\log z$ and $N_S^{-1}(z)=z\log z$, but this is not precise enough to prove that every large enough odd integer is the sum of two primes. You need far better approximations. Likewise, if $S$ is the set of squares, then $N_S(z) \sim \sqrt{z}$ and $N_S^{-1}(z)=z^2$, but this is not enough to prove that every large enough non-square integer is the sum of a square and a prime.

Is it possible to solve this problem using extremely precise approximations in all the asymptotic derivations discussed here? For instance, if $S$ is the set of prime numbers, then $N_S(z) \sim z/\log z$ and $N_S^{-1}(z)=z\log z$, but this is not precise enough to prove that every large enough even integer is the sum of two primes. You need far better approximations. Likewise, if $S$ is the set of squares, then $N_S(z) \sim \sqrt{z}$ and $N_S^{-1}(z)=z^2$, but this is not enough to prove that every large enough non-square integer is the sum of a square and a prime.

Maybe the following is true for sums of two primes and sums of a prime and and a square: there only finitely many $z$'s what can be expressed as $z=x+y$ in less than $k$ different ways, with $x\in S, y \in T$, regardless of $k$. This would imply that all but a finite number of $z$'s can be expressed as the sum in question.

Maybe the following is true for sums of two primes and sums of a prime and and a square: there only finitely many $z$'s that can be expressed as $z=x+y$ in less than $k$ different ways, with $x\in S, y \in T$, regardless of $k$. This would imply that all but a finite number of $z$'s can be expressed as the sum in question.

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Another question is whether my method is equivalent to the circle method.

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