# Questions tagged [goldbach-type-problems]

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### Possible rearrangments of double products containing sine function : [closed]

I know that the following question is not a fit (at all ) for this site , So , apologies ; but it interests me in very unusual way ; so I'm asking here . If not appropriate to post here tell me I'll ...
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### Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language

I thought of utilizing this lockdown period to study research papers in number theory by myself. I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ...
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### Can this number be interpreted as a fractal dimension?

Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$. Let's ...
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### Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture

Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that ...
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### How small can the smallest of the three “weak Goldbach” primes always be?

I've checked here for discussions of Helfgott's proof of the weak GC and found nothing that helps me with the following; apologies if I missed something. I'm probably being naive here (please ...
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### Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only sr+nm=2c } \\ 0 & \text{otherwise} \end{cases}$$ How does the "ugly"...
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### Which even numbers are known to be both prime gaps and the sum of 2 primes?

Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
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### Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, Hugh ...
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### Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds: i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...
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### Equivalence of Polignac to finite Goldbach?

Is Polignac's conjecture equivalent to a finite form of Goldbach? There is some discussion here as to the difference between Polignac & general Goldbach, but the similarity seems particularly ...
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### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. Let'...
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### Is $n = p-q$ equivalent to Goldbach's Conjecture?

One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.) Goldbach's conjecture states that every even integer greater than ...
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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 https://mathoverflow.net/questions/163211/upper-bound-for-r-...
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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 ...
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 $S\left(\alpha\right)=\underset{p\leq N}{... 1answer 209 views ### Tail of singular series of Goldbach problem Let$N$a large number and$P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$\underset{q>P}{\sum}\,\frac{\mu(q)^{2}}{\phi(q)^{2}}\overset{q}{\underset{a=1}{\sum}^{*}}e\... 0answers 222 views ### “Pseudo-random” subsets of additive bases We say that a subset B \subset \mathbb{N} is an (asymptotic) additive basis of order k if the set kB = B + \cdots + B = \mathbb{N} \setminus C, where C is a finite set of positive integers. ... 0answers 196 views ### Which upper bound for r_{0}(n) can be obtained through the Chinese Remainder theorem? Assume Goldbach's conjecture. Then for every integer n greater than one there exists a non negative integer r such that both n-r and n+r are prime numbers. For a given n, let's denote r_{0}(... 2answers 412 views ### Research on the structure of a non-Goldbach number? Has there been any research into the structure of a non-Goldbach number? This seems like it would be a profitable area for proof by contradiction, so I assume that someone has already done it. (i.e. ... 1answer 595 views ### A possible consequence of Dirichlet's theorem about primes in arithmetic progression EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained. "let's consider a composite natural number n greater or equal to 4. ... 0answers 482 views ### A conjecture on the relative size of Goldbach pairs? On leafing through some papers of John Nash (available online on his webpage) I found this intriguing little observation: Noticing that with larger even numbers it seemed to become possible to ... 0answers 414 views ### Divisor function inequality I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let z=N^{1/8}, then$$\sum_{w\leq z}\... 3answers 9k views ### About Goldbach's conjecture let's consider a composite natural number$n$greater or equal to$4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number$r$such as$(n-r)$and$(n+...
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 ...