Questions tagged [goldbach-type-problems]

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Reference request Re Vinogradov's ternary Goldbach proof

I believe that I.M. Vinogradov's proof of the ternary Goldbach conjecture used the observation that the number of ways $n$ can be written as a sum of three primes equals $$ \int_0^1 \sum_{p , q , r \...
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Does Cramer's random model of primes imply $L(n)w_{0}(n)=O(\log^{4}n)$?

Say $r$ is a Galois radius of $n$ of level $l=ab$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Denote by $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$, by $\...
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58 views

Duality between primality radius and Galois radius of maximal prime level

Say $r$ is a Galois radius of level $l=ab$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime, and let $\rho:=\frac{r}{l}$ the corresponding normalized Galois radius of $n$. Denote by $r_{l,k}(n)...
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45 views

Is the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ provable for some values of $a$, $b$ and $d$?

Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ ...
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76 views

Consecutive integers with consecutive primality radii

Disclaimer: the following observation is purely empirical and as such may not suit this website. Still, it may reveal interesting patterns about primes so I decided to ask this question nevertheless. ...
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117 views

Counting medium level Galois radii of an integer

This question builds upon About Goldbach's conjecture, whose beginning I copy-paste below, and upon Can a lower bound for this weakening of Goldbach's conjecture be reached?. "Let's ...
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2 votes
0 answers
151 views

A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?

The definitions are from these two questions: https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers https://math.stackexchange.com/questions/4349186/trying-to-understand-...
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0 votes
1 answer
93 views

$l$-th power radioprimal conjecture

I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$: $l$-th power radioprimal growth ...
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154 views

Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?

Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...
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143 views

Can a lower bound for this weakening of Goldbach's conjecture be reached?

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a weak primality radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, ...
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1 vote
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96 views

Tiling the set of integers with intervals of the form $[n-r_{0}(n),n+r_{0}(n)]$

Assuming Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ as well as $p_{\pm}(n):=n\pm r_{0}(n)$. Consider a sequence $(c_{m})_{m>0}$ defined by $c_{1}:=4$ and $c_{...
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2 votes
0 answers
82 views

$g$-gap radius of an integer

For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\...
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4 votes
3 answers
669 views

Consequences of Goldbach's Conjecture

In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement ...
3 votes
1 answer
202 views

Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$

This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$. Can one derive from Robin's criterion for RH an ...
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2 votes
0 answers
513 views

Weak Hardy-Littlewood-Goldbach conjecture

Assuming the weak Hardy-Littlewood-Goldbach conjecture as stated in this paper, does the density $d(\delta,\varepsilon)$ of integers $m$ below $n$ such that $$ \left\vert\frac{G(m)}{{\frak{S}}(m)m}-\...
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-3 votes
1 answer
258 views

Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?

This question is a follow-up to About Goldbach's conjecture and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ ...
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5 votes
1 answer
439 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 ...
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1 answer
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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 $...
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4 votes
1 answer
476 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 ...
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-1 votes
1 answer
215 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 ...
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-1 votes
1 answer
102 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 ...
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0 answers
68 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 ...
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1 vote
1 answer
284 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}-...
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3 votes
3 answers
708 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 $g(2n) \ge 1$ ...
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0 votes
0 answers
103 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,...
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1 vote
0 answers
71 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 ...
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3 votes
1 answer
272 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 ...
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3 votes
0 answers
238 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 ...
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0 votes
1 answer
376 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 $\chi$...
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1 vote
1 answer
352 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 ...
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0 votes
1 answer
595 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 ...
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0 answers
122 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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1 vote
0 answers
74 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 $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...
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2 votes
0 answers
132 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-...
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0 votes
2 answers
400 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 ...
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4 votes
0 answers
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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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-1 votes
1 answer
121 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 \...
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-2 votes
1 answer
194 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 ...
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0 answers
303 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 ...
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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 ...
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1 vote
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100 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}(...
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2 votes
0 answers
229 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 ...
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2 votes
0 answers
123 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 ...
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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-...
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-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{...
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2 votes
2 answers
980 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 ...
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0 votes
1 answer
142 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 ...
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-3 votes
1 answer
232 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 ...
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3 votes
0 answers
137 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 ...
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0 votes
1 answer
186 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 ...
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