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Questions tagged [goldbachs-conjecture]

For questions about Goldbach's conjecture: every even integer greater than two is the sum of two primes.

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Goldbach for certain classes of $n$

The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) states that In 1975, Hugh Montgomery and Robert Charles ...
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Implications of disproving the Goldbach's Conjecture

What would be the most important implications of finding an even number that cannot be expressed as the sum of two primes? Would the existence on one such number in anyway predict the likeliness of ...
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How many pairs $(x,y)$ such that $x+y=n$ have an $x$ or $y$ divisible by 3 but $x$ and $y$ are not equal to 3?

Let the set $S_{n}$ = {$(x,y):x,y \in \mathbb{O}$} such that $x+y=n$ where $\mathbb{O}$ is set of odd integers > 1. Let us define the function $f(n) = |S_{n}|$ that counts the number of pairs in $S_{...
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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"...
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Time complexity of finding the largest Goldbach partition

Suppose we are given a large even integer $N$, and we want to determine primes $p$ and $q$ such that $N = p + q$, subject to the conditions that $p \geqslant q$ and $p - q$ is as small as possible. (...
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$2n = \phi(a) + \phi(b)$

The values of the Euler phi function $\phi(n)$ are tabulated at OEIS A$000010$. Each of these values is even except for $\phi(1) = \phi(2) = 1$ . However, not every even number arises in this way. ...
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Useful reformulation of Goldbach's conjecture?

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$'th prime. Substituting $ y=\exp(\lambda x)$...
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Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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Weakening Goldbach hypothesis by allowing finitely many composites and 1

Is it an open question whether there is a finite set $N$ of positive integers such that for every positive even integer $n$ there are $n_1,n_2\in\mathbb P\cup N$ such that $n=n_1+n_2$? ($\mathbb P$ ...
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A heuristic argument for the Goldbach conjecture?

This question here is purely speculative so be warned if you read on: This question is related to a sequence $b_n$ which is defined here: A series related to prime numbers For the numbers $a_{2n,2}$ ...
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Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
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Is every positive integer greater than $2$ the sum of a prime and two squares?

I'm not sure if this conjecture is less hard than Goldbachs conjecture: any integer greater than $2$ is the sum of an odd prime and two squares of integers. Facts as: Every prime of the form $4n+1$ ...
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Is there any algorithm to identify the smallest example of gap size $n-1$ between consecutive prime numbers

In contemplating Goldbach's conjecture, I became interested in gaps between successive primes. If $n<a<b<2n$ and the range $a$ to $b$ is a primeless gap, then one could ignore any primes in ...
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Why can't we prove Goldbach's conjecture with this method?

1: Prime gap bounds: Consider the following non-asymptotic bounds for $\pi(x)$, proven by Dusart in 2018 (holding for $x>5393$): $$\frac{x}{\log(x)-1}<\pi(x)<\frac{x}{\log(x)-1.112}$$ To ...
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Two new conjectures related to Lemoine's and Goldbach's

Lemoine's conjecture can be written: $2n + 1 = p + 2q$ always has a solution in primes $p$ and $q$ (not necessarily distinct) for $n > 2$. It appears that the following very similar conjectures (...
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What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
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Lower bound for $g(n)$, the number of decompositions of 2n into ordered sums of two odd primes

I was coding an algorithm that calculates $g(n)$, the number of decomposition of 2n into ordered sums of two odd primes (A002372), or the number of Goldbach partitions. I noticed i can express the ...
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Can we hope for an elementary proof of a conjecture of Goldbach?

It is written on Wikipedia: "During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, ...
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Help in making Goldbach's conjecture reformulation rigourous?

Reformulation of Goldbach's conjecture Upon the suggestion of another stackexchange user this question has been reformulated to address the comments Useful reformulation of Goldbach's conjecture? ...
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Prime gap assuming Goldbach's conjecture

Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) $ the smallest primality radius of $n$ (non negative integer $r$ such that both $n-r$ and $n+r$ are primes) and by $r_{i+1}(n)$ the ...
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Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
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Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map $m\...
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Is every integer $\geq5$ the sum of two primes and a power of a prime?

Is every integer $\geq5$ the sum of two primes and a power of a prime (where $1$ is included in the prime powers)? I don't really expect someone to prove this here, but I wonder if the question has ...
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What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
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On 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 N}{...
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Why Goldbach Conjecture is difficult to solve in $\mathbb{R}[x]$ and $\mathbb{C}[x]$?

In an article on 'Comparing the close cousins $\mathbb{Z}$ and $\mathbb{F}_q[x]$', I've found the following The fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials ...
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Almost-norm restatement of Goldbach's conjecture.

Let $p_i$ be the $i$th prime number. Each number $x$ in $\Bbb{Z}$ can be expressed as a finite sum $\sum\limits_i (k_i p_i), \ k_i \in \Bbb{Z}$, in many ways. But define $\|x\| = \min\{ \sum_i |k_i| ...
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Best upper bound for $ r_{0}(n) $ under Goldbach and Chowla's conjectures

Assume Goldbach's conjecture. Then for any large enough composite integer $ n $ $ r_{0}(n) : =\inf\{r\ge 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ exists and is obviously smaller than $ n $ . Does the ...
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The Goldbach conjecture and sieves

Lemma: Any natural number less that $P_{i+1}^2$ is either prime or is divisible by one of $P_{1}, ..., P_{i}$. If we can show that any even natural number $2n$ which is in the interval $[P_{i}^2, P_{...
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Should Goldbach's Conjecture be restated thus? *Every integer $>3$ can be expressed as the average of two primes.*

Yes, every, not just even. If a number is the average (or difference) of two primes, by doubling the number it has a partition of those two primes. So, for example, $(7+31)/2=19$ becomes $7+31=2*19=...
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Related to proving Goldbach's Conjecture

How significant would it be to prove that every even number larger than $49$ can be written as the sum of two integers co-prime to $2, 3, 5,$ and $7$? I came across this result incidentally when ...
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A Generalization of Goldbach’s Conjecture

The Goldbach's conjecture is that $\forall n \in \mathbb{N}^*, \ \exists p,q \in \mathcal{P}$ such that $2n=p+q$. I wonder if there are some generalization giving more information about the ...
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Can you share some information to help study this unified sieve function for prime, twin prime and Goldbach sums of $2n$?

Let $p_i$ be the $i^{th}$ prime number. For Goldbach sums of $2n$, let $p_i$ be the largest prime less than $\sqrt{2n}$, define $$ P(p_i,n,x)=\sum_{p\le{p_i}}\frac{c_p}{p}\left(1+2\sum_{k=1}^{p-1}(1-\...
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Existence of a $G(x)$ that can generate all the even numbers?

Question This is a "spin-off" question of: Reformulation of Goldbach's Conjecture as optimization problem correct? I was wondering if a function existed such that: $$ G(x)^2 = (\sum_{i=1}^\...
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Does Linnik's approximation to Goldbach's problem also work for the power of 3, 5, 7, etc ?

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-...
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Equivalent conjecture to Goldbach's conjecture

I'm reading a paper regrading the basis orders. In that paper, I met with the following statement: $$3(\mathbb{P}\cup\{0 \})=\mathbb{Z}_{\geq 2}$$, Which, by definition, states that primes form ...
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For Riemann Hypothesis, many people seek physics intuition, why not for Goldbach Conjecture ?

All: As we all know, for Riemann Hypothesis research, many people seek physics intuition, to understand more fundamental reasons why Riemann Hypothesis shall hold. In this direction, we have ...
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Can we exploit the unique totient property of primes for a solution to Goldbach's Conjecture (strong)?

The following setup takes advantage of the fact that the totient of every prime p is p-1: Use an “All or nothing” approach in that: $$4\leq 2n=p+q\quad p,q\in\mathbb{P}\iff\forall 2n\geq 4, 2n=r+t\...
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Citizens and rebels: a twin prime related categorization of composites

Assuming Goldbach's conjecture and denoting by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$, consider the sequence $(u_n)_n$ such that $...
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Does the first Hardy-Littlewood conjecture imply $\sum\limits_{r=1}^{n-3}\Lambda(n-r)\Lambda(n+r)\sim 2C_{2}n$?

Hardy-Littlewood conjecture predicts that the number of Goldbach decompositions $p+q=2n$ should be asymptotically equal to $K\frac{n}{\log^2 n}\prod\limits_{p>2,p\mid n}\frac{p-1}{p-2}$ for a ...
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Lower bound for the ratio of primes and semiprimes as summands

Chen's theorem states that every large enough even integer is the sum of a prime and an almost prime, i.e. an integer that is either a prime or the product of two primes. As there are $s(n):=\pi(2n)-\...
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Does the assumption of both Goldbach and Elliott-Halberstam conjectures imply the twin prime conjecture?

Under Goldbach conjecture, denote by $ r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ for any large enough composite integer $ n $ . Say a positive composite integer $ n $ is hexahedral if ...
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Goldbach's conjecture and convergence of a Dirichlet series

Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) : =\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\} $. The assumption of GC implies $ r_{0}(n)<n $. Let's now consider the series $ G(s) : =\...
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Thoughts on Lehs conjecture $\forall n>2\in \mathbb N\exists a,b\in \mathbb N$ such that $a+b=n\land (a+ab+b)\in \mathbb P$. Lehs comet?

Lehs conjectured here that $\forall \ n>2\in \mathbb N,\exists\ a,b\in \mathbb N$ such that $a+b=n\land (a+ab+b)\in \mathbb P$. In comments, Crostul restated this as $\forall \ n\ge 4\in \mathbb N,...
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Where is the mistake: On the sum of two prime numbers.

Someone could help me find some error in the reasoning: We know, that the canonical decomposition of $n!$ is: $n!=\prod_{p_{i}\leq n}p_{i}^{\alpha_{i}(n)}$, where: $\alpha_{i}(n)=\sum_{t=1}^{r}[\...
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Maximal Goldbach Partitions?

A Goldbach partition $2n = p + q$ with $p$ and $q$ primes and $p \leqslant q$ is usually called minimal if the numbers $2n - k$ ($k = 1,\ldots, p-1$) are all composite. Reading through the literature,...
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Are those limits defined in terms of a positive integer actually independent thereon?

Under Goldbach's conjecture, denote by $ r_{0}(n)=\inf\{r\gt 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and by $ k_{0}(n)=\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $ for $ n $ large enough. Say $ n $ is $ k $ -...
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An equivalent form of the Goldbach conjecture using the radical of an integer and the Euler's totient function, and a related problem

Let $\phi(n)$ denoting the Euler's totient function and $\operatorname{rad}(n)$ the so-called radical of an integer (see this Wikipedia.) The MathWorld's article dedicated to the Goldbach conjecture ...
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About Goldbach's conjecture : a certain differential equation

This question is related to https://mathoverflow.net/questions/61842/about-goldbachs-conjecture Let $ y(x) $ be a function such that $ \alpha_{n}=O(y(n)) $ . taking $ y(x) : =x^{1/2}\log^{2} x $ ...
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Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} =...