# Questions tagged [goldbach-type-problems]

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72
questions

**-6**

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124 views

### Sieve methods and primal height

Define the primal order $\omega_{\mathbb{P}}(n)$ of an integer $n$ as the smallest $k$ such that the $k$-th iterate of the prime counting function $\pi$ evaluated at $n$ is composite and the primal ...

**-1**

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**0**answers

95 views

### Gap repeating integers

Under Goldbach's conjecture, set $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$. Define similarly $r_{i+1}(n):=\inf\{r>r_{i}(n),(n-r,n+r)\in\...

**-1**

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**0**answers

152 views

### Does one have $\alpha_{n}\ll\sum_{p^m\leq n, m\geq 2}\Lambda(p^{m})\log n$?

This question is a follow-up to my question "About Goldbach's conjecture" (direct link: About Goldbach's conjecture) whose beginning I copy-paste below:
"Let's consider a composite ...

**5**

votes

**1**answer

339 views

### Weak Goldbach conjecture with distinct primes for odd integers between $4\times 10^{18}$ and $10^{27}$

This is related to the conjecture that all odd integers greater than $17$ can be written as the sum of 3 distinct primes.
Schinzel showed that the Goldbach conjecture implied this in 1959 and as the ...

**0**

votes

**1**answer

123 views

### Upper bound for the number of $k$-central numbers in a prime gap

Let $I_{n}:=]p_{n},p_{n+1}[$ be the open interval between the $n$-th and $(n+1)$-th prime. Under Goldbach's conjecture, denote by $r_{0}(m)$ the smallest positive integer $r$ such that both $m-r$ and $...

**0**

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**0**answers

74 views

### Primality radius and Hamming distance

This question is a follow up to About Goldbach's conjecture, whose relevant content I copy paste below so as to make it self-contained:
"let's consider a composite natural number $n$ greater or ...

**0**

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**0**answers

56 views

### Gaussian primality radius

I'm trying to generalize the notion of primality radius of a positive composite integer $n$, defined as a positive integer $r$ such that both $n-r$ and $n+r$ are prime, to Gaussian integers.
As such, ...

**3**

votes

**0**answers

339 views

### Staircase numbers

Assuming Goldbach's conjecture, denote as usual by $r_{0}(n)$ for any large enough positive integer $n$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime.
Let's define the ...

**-1**

votes

**1**answer

197 views

### A number theoretical identity of exponential sum

I try to understand a number theoretical identity used by
Jan-Christoph Schlage-Puchta in this answer.
He defined the function
$$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$
where $\Lambda(n)$ is ...

**0**

votes

**1**answer

96 views

### Does one have $2r_{0}(n)\lesssim k_{0}(n)(\log n)^{1+1/k_{0}(n)}$?

Under Goldbach's conjecture, I'm trying to find an upper bound for $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ that would generalize Cramer's conjecture.
Denoting by $k_{0}(n)$ the quantity ...

**0**

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61 views

### Order of growth of the error term of the log-exponent of the average prime gap

Disclaimer: I might have already asked this question or a very similar one but couldn't find it if it is so. Hope it will be judged somehow interesting anyway.
Assuming Goldbach's conjecture, let's ...

**0**

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**0**answers

51 views

### Asymptotics for the $m$-th integer of given fundamental primality radius and order of centrality

Assuming Goldbach's conjecture, define the "fundamental primality radius of $n$", denoted by $r_{0}(n)$, as $\inf\{r>0\mid (n-r,n+r)\in\mathbb{P}^{2}\}$, and the "order of centrality ...

**0**

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**0**answers

77 views

### Prime constellation containing the primes in $[n-r_{0}(n),n+r_{0}(n)]$

Assume throughout this question both Goldbach's and Hardy-Littlewood k-tuples conjecture. Then for any sufficiently large positive composite integer $n$ the quantity $r_{0}(n):=\inf\{r>0,(n-r,n+r)\...

**1**

vote

**1**answer

255 views

### Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...

**1**

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166 views

### How Important was Estermann's Theorem : Almost all even positive integers are sum of two primes and Why?

I am a second year masters student and earlier this year I was reading the research paper : "On Goldbach's Problem : Proof That Almost All Even Positive Integers are Sums of Two Primes" By T....

**3**

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**3**answers

684 views

### Goldbach conjecture and the representation number

Let $g(2n)$ be the number of representations of $2n=p+q$ with primes $p$ and $q$. Many people have asked whether $g(2n) \ge 2$ when $2n = p+q$ for some primes $p$ and $q$. That is, does $g(2n) \ge 1$ ...

**0**

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99 views

### Fundamental primal polynomial associated to an integer

Under Goldbach's conjecture, let $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ for any composite positive integer $n$. Let also $g_{1}(n),\cdots,...

**1**

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70 views

### Projection of cocyclic Gaussian primes on the real axis

I just stumbled upon https://math.stackexchange.com/questions/2372062/4-concylic-points-of-gaussian-primes after a quick Google search about cocyclic Gaussian primes.
As I've been investigating about ...

**3**

votes

**1**answer

249 views

### Prime gap distribution in residue classes and Goldbach-type conjectures

Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question).
The general problem that I try to solve is ...

**3**

votes

**0**answers

225 views

### A nice pattern about Goldbach conjecture in French Wikipedia

In the following link: https://fr.m.wikipedia.org/wiki/Conjecture_de_Goldbach, one can see a nice pattern of pink and blue lines coming from each prime number, the intersection points thereof are ...

**1**

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**1**answer

327 views

### Congruential equidistribution, prime numbers, and Goldbach conjecture

Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$...

**1**

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**1**answer

331 views

### Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions:
$N_S(z)$ is asymptotic continuous version of the function counting the ...

**0**

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**1**answer

551 views

### Paradox in additive combinatorics

Let $S$ be an infinite set of positive integers. Let us define the following quantities:
$N_S(z)$ is the number of elements of $S$, less or equal to $z$
$r_S(z)$ if the number of positive integer ...

**0**

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112 views

### General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...

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70 views

### $t$-balanced numbers

Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...

**2**

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128 views

### Is this conjecture equivalent to Polignac's conjecture?

Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...

**0**

votes

**2**answers

353 views

### 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 ...

**3**

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**0**answers

104 views

### 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 ...

**-1**

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**1**answer

119 views

### Statement about upper density of even numbers satisfying the Goldbach condition

For $A\subseteq \mathbb{N}$, let the upper density of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Let $$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \...

**-2**

votes

**1**answer

187 views

### Are there infinitely many karmic numbers, i.e numbers whose primality radii equal one or a prime power?

For $n$ a large enough positive composite integer, say $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime. Say $n$ is a karmic number if the following holds: $r$ is a primality radius ...

**0**

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268 views

### Equivalent Statements of Goldbach Conjecture in Terms of the Properties of Riemann Zeta Function?

Riemann Hypothesis has many equivalent statements.
Many of them are not about prime distribution, instead, are about the properties of Riemann Zeta function, such as the distribution of zeros of Zeta ...

**3**

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219 views

### Riemann hypothesis and ternary Goldbach

Is there any result of the following shape: There exists an absolute constant $\delta>0$ such that the Riemann hypothesis for some $L$-functions is equivalent to the following estimate for all ...

**1**

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97 views

### Upper bound for $\alpha_{n}$ from Mertens' third theorem

This question is a follow-up to About Goldbach's conjecture.
I would like to know if an unconditional upper bound for $\alpha_{n}$, defined as $n(N_{2}(n)-\dfrac{nN_{1}(n)}{P(n)})$ (where $N_{2}(...

**2**

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**0**answers

203 views

### Distribution of Goldbach's weak-conjecture's prime-triples

From Harald Helfgott's proof of Goldbach's weak conjecture,1
we know that every odd number $> 7$ is the sum of three odd primes.
If $n$ is such an odd number, say that two sums that yield $n$
are ...

**2**

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**0**answers

122 views

### Primality radii and Sidon sets

I learned tonight what a Sidon set is, in a book about Erdős. This notion inspires me the following question :
For $n$ a large enough composite integer, say $r>0$ is a primality radius of $n$ if ...

**0**

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**0**answers

61 views

### Is the fundamental partition associated to $n$ the partition of $r_{0}(n)$ in $k_{0}(n)$ parts that maximizes entropy?

As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-...

**-2**

votes

**1**answer

71 views

### Approximation for $ \inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ by minimizing a distance

Under Goldbach's conjecture, let $r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and $k_{0}(n) : =\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $. The PNT implies that one can expect to have $ \dfrac{...

**2**

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**2**answers

901 views

### If Goldbach's Conjecture is eventually true, is it necessarily true?

We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite ...

**0**

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**1**answer

137 views

### Sergei numbers : even integers n being a prime gap at least n times

Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...

**-3**

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**1**answer

222 views

### Can this weakening of Polignac's conjecture be proven?

Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...

**3**

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135 views

### Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?

In this previous question of mine
I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture ...

**0**

votes

**1**answer

180 views

### Upper bound for $\sum r_{0}(n)$

Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound ...

**4**

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**0**answers

248 views

### Error term for Vinogradov's three prime theorem

It can be shown that $$\sum_{a + b + c = N}\Lambda(a)\Lambda(b)\Lambda(c) = \frac{1}{2}\mathfrak{S}(N)N^2 + O(N^2\log^{-A} N)$$ for some $1\ll \mathfrak S(N)\ll 1$using the circle method. Are there ...

**12**

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361 views

### On a relaxed form of Goldbach's conjecture proposed by Erdős

The Goldbach's conjecture says that:
"Every even integer greater that $2$ is the sum of two prime numbers".
Let $\varphi$ denote the Euler's totient function. I remember that a long time ago I read ...

**13**

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**0**answers

482 views

### Is every powerful number the sum of a powerful number and a prime?

A positive integer $n$ is called powerful (OEIS: A001694) if $p^2$ divides $n$ whenever $p$ is a prime that divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive ...

**5**

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**1**answer

628 views

### The prime gap 2 and the prime gap 4, are they equally common?

This question was posed today by a student of mine, and I have not been able to find any relevant references.
Let $p_1, p_2, p_3, \ldots$ be the sequence of prime numbers listed in increasing order, ...

**2**

votes

**1**answer

173 views

### Upper bound for an exponential sum in Waring-Goldbach problem

In Waring's problem, we have Hua's estimate
$$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$
where $(a,q)=1$.
?Do you know a similar upper bound for the sum
$$...

**0**

votes

**1**answer

304 views

### 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 ...

**6**

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**1**answer

342 views

### 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 ...

**0**

votes

**1**answer

377 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 "ugly"...