Questions tagged [goldbach-type-problems]

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9 votes
1 answer
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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 ...
2 votes
0 answers
82 views

Bateman-Horn-type generalization of the Goldbach conjecture

The Bateman-Horn conjecture is a generalization of the twin prime conjecture that roughly states that given a set $S=\{f_1, \dots, f_m\}$ of irreducible polynomials with integer coefficients, there ...
4 votes
0 answers
159 views

Effective bound for odd numbers expressed as sums of three primes

I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
3 votes
2 answers
629 views

Goldbach conjecture and the difference of two primes

The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes? That is, if $2n$ is a sum of ...
0 votes
0 answers
366 views

Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture

I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight ...
4 votes
1 answer
546 views

Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is hereafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, Hugh ...
8 votes
1 answer
893 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) $ ...
-1 votes
1 answer
230 views

Inversion shift of a Galois radius

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ ...
6 votes
1 answer
688 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 ...
2 votes
0 answers
199 views

Is the Goldbach conjecture easier if we allow 1 as a prime?

I hope this is the right site for the question. Is the Goldbach conjecture easier if we allow 1 as a prime? (12=1+11 would be allowed as Goldbach sum for 12) IOW: if we can prove Goldbach for the case ...
8 votes
1 answer
1k views

Number field analogue of the Goldbach Conjecture

Is there a generalization of Goldbachs conjecture for prime ideals in number fields?
4 votes
1 answer
763 views

On Buchstab et al's "forgotten" sieve and the Goldbach conjecture for certain integers

There is a somewhat forgotten sieve-theoretic approach to the Goldbach conjecture, due to Buchstab et al, see e.g. pp.247-248 of R.D. James. On p.247, James defines some function $F$ such that for any ...
1 vote
0 answers
156 views

Another Goldbach variation for odd numbers?

Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime. Dabbling in the dark art ...
0 votes
0 answers
83 views

Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary

Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...
3 votes
0 answers
239 views

Lower bounding the number of Galois radii of an integer

Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$. Does it suffice to ...
10 votes
1 answer
2k views

Current status of Waring-Goldbach problem

Is the following statement proved? For any positive integer $k$ there exists positive integer $n$ such that all sufficiently large integers may be represented as $p_1^k+p_2^k+\dots+p_n^k$ for primes $...
1 vote
0 answers
283 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 Galois radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, where $\...
3 votes
1 answer
709 views

Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
0 votes
0 answers
152 views

$k$-Taiwan numbers

Say a positive composite integer $n$ is a $k$-Taiwan number if $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}=p^{k}$ for some prime number $p$, and a Taiwan number if it is a $k$-Taiwan number ...
2 votes
0 answers
242 views

Selberg's 1943 result on primes in short intervals and primality radius

This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...
5 votes
1 answer
585 views

Why does this convolution of the prime counting function $\pi$ look like a parabola?

In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture: $$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$ The ...
0 votes
0 answers
152 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^{...
5 votes
3 answers
1k 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 ...
2 votes
0 answers
68 views

Prime powers gap of type $(a,b)$

For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime. Denote by $PPG_{a,b}(m)$ ...
3 votes
0 answers
211 views

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 \...
2 votes
0 answers
51 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 $l$ ...
2 votes
0 answers
547 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}-\...
-3 votes
1 answer
266 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 $n-r$ ...
0 votes
0 answers
166 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-...
2 votes
0 answers
257 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-...
0 votes
1 answer
94 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 ...
1 vote
0 answers
103 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_{...
2 votes
0 answers
84 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+\...
3 votes
1 answer
227 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 ...
4 votes
1 answer
544 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 ...
34 votes
3 answers
11k 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 $(n-r)$ ...
0 votes
1 answer
144 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 $...
4 votes
3 answers
765 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 $g(2n) \ge 1$ ...
12 votes
0 answers
554 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 ...
-1 votes
1 answer
236 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 ...
-1 votes
1 answer
108 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 votes
0 answers
73 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 ...
1 vote
1 answer
311 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}-...
0 votes
0 answers
118 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 vote
0 answers
75 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 ...
0 votes
2 answers
499 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 votes
1 answer
339 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 ...
0 votes
1 answer
451 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 $\chi$...
3 votes
0 answers
254 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 vote
1 answer
377 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 ...