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155 views

Any heuristics explaining why one seems to have $2n=p+q\Rightarrow\pi(p)+\pi(q)=Li(2n)+o(1)$?

The title explains almost everything: suppose $2n$ is a Goldbach number, i.e. an even positive integer such that there exist two primes $p$ and $q$ fulfilling $2n=p+q$. It seems that "most of the ...
1
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0answers
111 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [closed]

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...
1
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1answer
287 views

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 ...
0
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2answers
448 views

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 ...
4
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1answer
205 views

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, ...
0
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0answers
91 views

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 ...
2
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1answer
126 views

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 ...
4
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0answers
355 views

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. ...
8
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3answers
645 views

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 ...
0
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0answers
154 views

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 ...
7
votes
1answer
576 views

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) $ ...
0
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2answers
257 views

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 ...
0
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1answer
330 views

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 ...
2
votes
1answer
334 views

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 ...
7
votes
1answer
706 views

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 ...
1
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0answers
223 views

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 ...
0
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1answer
172 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 $$ ...
4
votes
1answer
172 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. ...
1
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0answers
134 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 ...
1
vote
2answers
377 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. ...
0
votes
1answer
491 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 $4$. ...
5
votes
0answers
435 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 ...
0
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0answers
325 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 ...
4
votes
1answer
461 views

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 ...