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### Estimating the number of twin primes of given natural configuration order

This question is a follow-up from About Goldbach's conjecture. Let
$$\mathrm{Co}_{k}(x):=\{n\le x:\mathrm{ord}_{c}(n):=\pi(\sqrt{2n-3})=k\},~~~\mathrm{co}_{k}(x):=\vert\mathrm{Co}_{k}(x)\vert$$
...

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### Does $\pi(n+r)+\pi(n-r)$ decrease as $r$ increases?

Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a ...

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### Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...

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### Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...

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### Is a certain sumset derived from primes of a certain form the set of all naturals?

OEIS A167055 Numbers n such that $12n + 5$ is prime.
$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$.
I conjecture that the set of the sum of every two items of this ...

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### Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture?

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 ...

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### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

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### Average order and upper bound of $r_{0}(n)$

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non-negative integer $r$ such that $n-r$ and $n+r$ are both primes. For a given $n>1$, the smallest such $r$ will be ...

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### An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...

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### Tail of singular series of Goldbach problem

Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ ...

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### “Pseudo-random” subsets of additive bases

We say that a subset $B \subset \mathbb{N}$ is an (asymptotic) additive basis of order $k$ if the set $kB = B + \cdots + B = \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. ...

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### Which upper bound for $r_{0}(n)$ can be obtained through the Chinese Remainder theorem?

Assume Goldbach's conjecture. Then for every integer $n$ greater than one there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are prime numbers. For a given $n$, let's denote ...

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### Research on the structure of a non-Goldbach number?

Has there been any research into the structure of a non-Goldbach number? This seems like it would be a profitable area for proof by contradiction, so I assume that someone has already done it. (i.e. ...

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### A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained.
"let's consider a composite natural number $n$ greater or equal to $4$. ...

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### A conjecture on the relative size of Goldbach pairs?

On leafing through some papers of John Nash (available online on his webpage) I found this intriguing little observation:
Noticing that with larger even numbers it seemed to become
possible to ...

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### Divisor function inequality

I have been reading a paper on the Goldbach conjecture found at
http://people.exeter.ac.uk/pt224/Goldbach.pdf.
At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then
$$\sum_{w\leq ...

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### The minimal Goldbach basis

Let $n \in \mathbb{N}, n \geq 2$. By minimal Goldbach basis $G_{2n}$(if it is nonempty) of $2n$ , I mean the minimal set of primes such that every even number less than or equal to $2n$ can be written ...