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The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in Upper bound for $r_{0}(n)$ through probabilities that seems to show that the former allows to get an upper bound for the quantity $r_{0}(n)$ of the form $r_{0}(n)=O(\log^{2}n)$ which implies both Cramer's conjecture and asymptotic Goldbach's conjecture (since Goldbach's conjecture is equivalent to $r_{0}(n)<n$ for all $n>1$). But are there rigorous published results that establish such an implication?
Thanks in advance.

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up vote 6 down vote accepted

I don't believe so. It's generally believed that any proof of the Hardy-Littlewood $k$-tuple conjecture (even for $k=2$) would use methods that could quickly be adapted to the Goldbach conjecture, but that's not the same as a rigorous deduction.

Writing a particular even integer $N$ as the sum of two primes is the same as looking for simultaneous prime values of the polynomials $n$ and $N-n$. Hardy-Littlewood gives asymptotics for the number of $n\le x$ for which both are prime (for this to hold, we need to consider negative of primes as primes as well). But the Goldbach conjecture isn't concerned with $x$ going to infinity; it's concerned with $x=N$. So one would have to conjecture the size of the error term in the Hardy-Littlewood asymptotic (including dependence on the coefficients of the linear polynomials) to derive Goldbach this way.

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