Questions tagged [goldbach-type-problems]

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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 ...
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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\...
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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
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1answer
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 ...
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1answer
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 $...
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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 ...
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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, ...
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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 ...
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1answer
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 ...
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1answer
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 ...
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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 ...
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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 ...
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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)\...
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1answer
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}-...
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0answers
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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3answers
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 $g(2n) \ge 1$ ...
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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,...
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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
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1answer
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
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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 ...
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1answer
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 $\chi$...
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1answer
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 ...
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1answer
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 ...
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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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0answers
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-...
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2answers
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 ...
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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 ...
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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 \...
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1answer
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 ...
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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 ...
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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 ...
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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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0answers
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 ...
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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 ...
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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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1answer
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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2answers
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 ...
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
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0answers
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 ...
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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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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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1answer
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
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1answer
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 $$...
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1answer
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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1answer
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
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1answer
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"...