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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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Is it possible to have this overlap between Goldbach and the twin prime conjectures?

This question is related to this. But, here it is related Goldbach's conjecture. Any even number greater than $4$ is the result of addition of two prime numbers one of which is the lower of a twin ...
Zuhair's user avatar
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Every even number is the sum of at most three primes

I'm failing to find online references to the following problem, which to me seems a slight weakening of the Goldbach conjecture. Conjecture: every even integer $n$ is the sum of at most three primes. ...
CryptoZiddy's user avatar
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1 answer
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If $|G_{2n}| \geq 2$ for all $n \geq 4$, does it imply Goldbach's Conjecture? My Conjecture onto proving Goldbachs Conjecture. [closed]

Let's look some definitions before we start. Let $n \in \mathbb{Z}^+$ and $G_n := \{\text{primes } p : p \nmid n \, \wedge \, p<n\}$. Let $U(n)$ be group of units of the cyclic group $\mathbb{Z}/n\...
Joshua Ortiz's user avatar
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Explaining the irregularities of the number of Goldbach pairs

I am working from a paper by Hardy and Littlewood from 1923 which attempts to construct an approximation to the number of Goldbach pairs for a given $n$. On page 32, they present a product which ...
Goldbug's user avatar
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question on estimator for $\frac{\pi(n)}{n}$ and $\frac{\pi_2(n)}{\pi(n)}$

$\pi(n)$ and $\pi_2(n)$ represent the count of primes and count of twin primes $\leq n$ respectively. Suppose we want to estimate $\frac{\pi(n)}{n}$. One way which obviously is not error-free is to ...
sku's user avatar
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Sum of two values in the range of $\sigma_1(n)$

$\mathcal{Q}$ : Is it true that for $n>3$, there exists $u$ and $v$ in $\mathbb N$ such that $$n=\sigma_1(u)+\sigma_1(v),$$ where $\sigma_1(k)$ is the sum of the positive integer divisors of $k$ ...
Eugen Ionascu's user avatar
2 votes
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What's the composition of primes in Goldbach's Conjecture?

Lemma For $n>5$, every even number $2n$ can be expressed as a sum of four primes $p_0 + p_1 + p_2 + p_3$. Proof Let $p_0$ be an odd prime and $m = 2n - p_0$ an odd number. Applying the Weak ...
vengy's user avatar
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Plot of the ratios of Goldbach pairs

Preface I was playing around with matplotlib to generate some number sequences. I wound up looking at Goldbach pairs and manipulating them in different ways. End result was the following plots. I can'...
Mudsy's user avatar
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Asymptotic behaviour of infinite sum over prime powers

I am currently studying analytic number theory and my teacher suggested to ask here if the following sum $S = \sum_{p} x^p = x^2 + x^3 + x^5 + x^7 + ...$ Where $p$ is a prime number is known and if it ...
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Is any power of a theta function over primes a modular form?

Apologies for the vague title but I wasn't sure how to word this. To preface my question, let's recall the theta function: $$\theta(\tau) = \sum_{n \in \mathbb Z} e^{i \pi n^2 \tau}$$ This function is ...
GaseousButter's user avatar
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Is every semiprime the difference of two coprime squares of opposite parity?

If this holds for all $n\gt3$ then we have another way of expressing Goldbach's conjecture as a product of two primes because the sum of the factors of the difference of two squares equals $2n$: $n^2-...
M. B. Jones's user avatar
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1 answer
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Number of even integers not satisfying Goldbach's conjecture from Vinogradov. Infinity of numbers not satisfying the Goldbach's conjecture.

If $A(x)$ is the number of even integers less than $x$ that don't write as a sum of two (odd) primes, then $$ \lim_{x\to \infty} \frac{A(x)}{x} = 0$$ That is what is written in my book (Elementary ...
niobium's user avatar
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2 answers
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Can we use the proof of the weak Goldbach conjecture to also prove the strong Goldbach conjecture?

Why doesn't proof of the weak Goldbach conjecture also prove the strong Goldbach conjecture? Actually I am referring to this link. My question is why the logic used in this question cannot be used ...
Ok-Virus2237's user avatar
3 votes
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Is the weak Goldbach conjecture proved? [duplicate]

The Wikipedia page of the Goldbach's weak conjecture states that "In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the ...
Ok-Virus2237's user avatar
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Is the sequence $p_n-n+1$ related to the Goldbach conjecture via the Dirichlet inverse of of the Euler totient?

I am trying to learn what the Goldbach conjecture is and I therefore ran this Mathematica program where I tried to incorporate the conjecture: ...
Mats Granvik's user avatar
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2 votes
2 answers
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Is the stronger form of Dirichlet's theorem on arithmetic progressions strong enough to prove Goldbach's (asymptotic) conjecture?

The stronger form of Dirichlet's conjecture states that, for example, $$\lim_{N\to\infty} \frac{\text{ the number of primes } \leq N \text{ of the form } 1+8k }{\text{ the number of primes } \leq N \...
Adam Rubinson's user avatar
1 vote
1 answer
111 views

Can this light variant of Goldbach be proven?

Goldbach's conjecture states that every even integer greater than $2$ is the sum of two (not necessarily distinct) prime numbers. It seems that for $n>6$ , we can choose a representation with ...
Peter's user avatar
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Is the function for the Weak Goldbach Conjecture an increasing function?

Premise There is a function used to count the number of ways a given odd integer larger than 5 can be written as the sum of three prime numbers. I have seen the function ($f_{3}$) expressed as the ...
tkellehe's user avatar
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If an arithmetic function is multiplicative, non-zero at a prime, and "prime-linear", is it the identity?

Let $f:\mathbb{N}\to\mathbb{N}\cup\{0\}$ be a function. Let $f(1)=1,$ and $f(ab)=f(a)f(b)$ whenever $\gcd(a,b)=1.$ Note that I am assuming that $f$ is multiplicative but not completely multiplicative. ...
aqualubix's user avatar
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Why is the function $r(n)$ of particular interest in Circle method?

I was reading Goldbach problem(Ternary version) and encountered the Hardy-Littlewood circle method.In this method,we work with a number $r(n)=\sum\limits_{n=n_1+n_2+n_3}\Lambda(n_1)\Lambda(n_2)\Lambda(...
Kishalay Sarkar's user avatar
2 votes
1 answer
206 views

Goldbach's conjecture for odd numbers satisfying prime relations

Inspired by this question and remembering Bertrand's postulate, I wondered. Does any set of random odd numbers satisfying Bertrand's postulate satisfy Goldbach's conjecture (i.e. that every even ...
jorisperrenet's user avatar
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120 views

Hypothesis to split an even number as the sum of two primes as per Goldbach's Conjecture

I have the following conjecture in regard to Goldbach's Conjecture which I have found via my own experimentation. I wanted to run it by here to see if it is correct and can be proved formally. $\...
Pathlessbark8's user avatar
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Smaller Bound to find primes for Goldbach's Conjecture

I am an undergraduate student studying mathematics and have come across an observation. I thought this would be a great place to discuss it. In my attempt to understand Goldbach's Conjecture in a ...
Pathlessbark8's user avatar
2 votes
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91 views

A group theoretic formulation for Goldbach's conjecture

Consider the action of multiplying by $(n-1) \pmod n$, in the group of units $U(n)$ when $n$ is an even number. This action is a map $f$ defined on $U(n)$ such that for each $x \in U(n), f(x) = (n-1)x ...
Márcio Palmares's user avatar
1 vote
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Goldbach Conjecture-Does the complete graph of odd primes up to 2n with averages as edge labels contain all integers up to $p_{\pi(2n)}$?

Define $G_n=K_{\pi(2n)-1}\circ C_1 $ to be the complete graph on $\pi(2n)-1$ vertices composed with the self-loop to yield a complete graph with self loop edges and where $\pi(k)$ is the prime ...
Locke Demosthenes's user avatar
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Goldbach's Conjecture: Counterexample of "necessary condition"

Definitions: Divisor Function: $$\sigma_x(n) = \sum_{d\mid n} d^x$$ Euler's Totient Function: $$\phi(n) = \# \{m \in \mathbb{Z}^+ \mid (\gcd(m,n)=1) \wedge (1 \le m \le n)\}$$ Conjecture: The ...
Joshua Ortiz's user avatar
2 votes
1 answer
117 views

Proving a variation of Lemoine's Conjecture by assuming the strong Goldbach Conjecture

In 2013, when I was just a totally newbie recreational mathematician, I read about Levy's conjecture (i.e., Lemoine's conjecture, stating that all odd integers greater than 5 can be represented as the ...
Marco Ripà's user avatar
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Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.

Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
Marco Ripà's user avatar
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5 votes
1 answer
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An "almost all" result for the binary Goldbach problem

I have a question. My professor in the lecture said that Vinogradov's method by applying the Hardy-Littlewood circle method (minor and major arc) for the ternary Goldbach problem can be used to prove ...
3m0o's user avatar
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1 vote
1 answer
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Equivalences of Goldbach conjecture - Added value?

Let $I(n)$ be the set of the first $n$ odd integers, $S_p(n)$ be a subset of the last $p$ elements of $I(n)$ such that $p$ is some prime number, and $a_{n-\left(\frac{p-1}{2}\right)}$ the $n-\left(\...
Juan Moreno's user avatar
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3 votes
1 answer
181 views

Are there any large numbers found that seemed eerily close to disproving Goldbach's conjecture?

This question is about any numbers that seemed unusually close to disproving Goldbach's conjecture. Meaning any large numbers (say above 100) that had very few sets of primes which satisfied the ...
Juel Herbranson's user avatar
4 votes
0 answers
260 views

Can any positive even number be expressed as an XOR of two prime numbers?

I just came up with this question when I was thinking about the Goldbach conjecture. For example, $$2=5 \oplus 7$$ $$4=3 \oplus 7$$ $$6=3 \oplus 5$$ $$8=3 \oplus 11$$ $$10=7 \oplus 13$$ $$12=7 \oplus ...
SegmentTree's user avatar
1 vote
0 answers
54 views

Is it true that for sequences that satisfy the property-type in Goldbach's conjecture, there is an integer which cannot be expressed in a unique way?

Let $A\subset \mathbb{Z}$ be such that $\exists\ c\in\mathbb{Z}$ such that $\forall\ n\geq c: \exists\ a,a'\in A$ with $a+a' = n.$ In other words, $A$ satisfies sort-of Goldbach conjecture, but for ...
Adam Rubinson's user avatar
3 votes
0 answers
110 views

Is every large enough odd integer the sum of a prime or prime power and a power of $2$?

I've been thinking for some months about a slightly weaker form of Goldbach's conjecture: namely that every large enough integer is the sum of two prime powers or primes, that is $\exists C>0,\...
Sylvain Julien's user avatar
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1 answer
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Why is this infinite series _intuitionistically_ Cauchy?

I'm currently writing a short paper on Intuitionism for uni. The subject of this paper is the decay of the intermediate value theorem under intuitionism. I have found a proof for this but I have a ...
Jord van Eldik's user avatar
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1 answer
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If Goldbach's conjecture is false, is it possible that there are only a finite number of failing cases?

If Goldbach's conjecture is false, is it possible that there are only a finite number of failing cases? I know it is probably unknown, but any reference to something addressing this question would be ...
François Huppé's user avatar
2 votes
1 answer
68 views

Why does $\sum\omega\left(2n-p\right)=\sum\pi_{p,b}\left(2n-p\right)$?

Why does: $$ \sum_{_{3\leq p\leq2n-3}}\omega\left(2n-p\right)=\sum_{_{3\leq p\leq2n-3}}\pi_{p,b}\left(2n-p\right) $$ where: $ p\text{ is prime} $, $ \omega\left(x\right)\text{ counts each distinct ...
François Huppé's user avatar
6 votes
1 answer
232 views

On an approximation to Goldbach's conjecture

I've been recently reading Yuan Wang's paper on an approximation to Goldbach's problem, in which he showed that Proposition 1: For all large even integer $x$, there exists $1<n<x-1$ such that $n(...
TravorLZH's user avatar
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2 votes
0 answers
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Primal entanglement

Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. Say two composite integers $a$ and $b$ are primally entangled if $r_{0}...
Sylvain Julien's user avatar
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1 answer
146 views

Why can't we prove Goldbach’s Conjecture with modular 2?

"Every even whole number greater than 2 is the sum of two prime numbers." $2n = p_1 + p_2\hspace{1cm} ,n>1 \\ 2n = [(p_1-1) +1] + [(p_2-1) +1] \\ 2n = (p_1-1) + (p_2-1) + 2 \\ 0 \equiv 0 \...
Jax's user avatar
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0 answers
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Powers of $2$ and $3$ and primality radius

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime. Are there infinitely many such couples $(n,r)$ of the form $(p^a,q^b)$ with positive $a$ and $b$ and $\{p,...
Sylvain Julien's user avatar
2 votes
0 answers
113 views

Odd semiprimes as differences of two even perfect squares, divided by 4 - consequences regarding Goldbach's Conjecture?

My observation is that every odd semiprime can be written as the difference of two even perfect squares, divided by 4, or, in other words, in order to locate--and possibly factorize--odd semiprimes, ...
Jesko Matthes's user avatar
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0 answers
54 views

Does Goldbach's Conjecture hold true for other conditions?

I have been reading up on Goldbach's conjecture and how I understand it is as follows: For all values of x that satisfy x % 2 == 0, where x is an element from the set of natural numbers starting at 4, ...
Curious_Student's user avatar
-1 votes
1 answer
62 views

Properties of the even number that doesn't satisfy the Golbach's Conjecture. [closed]

This is a little vague question, but I think this is the best place to ask it. We haven't found an even number which cannot be written as a sum of two primes, but mathematicians must have studied that ...
Rounak Sarkar's user avatar
6 votes
1 answer
161 views

Is it possible that every $P$ as a prime number, can be expressed as a prime factor of $E$ such that $E$ is the sum of a pair of twin primes?

Curious about the Goldbach conjecture, and reading about twin primes, I was wondering if it is possible that every prime number as $P$, can be expressed as a prime factor of at least one $E$ such that ...
Isaac Brenig's user avatar
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1 vote
1 answer
267 views

Equivalence between Goldbach and Euler's statements

We know that Goldbach wrote Euler, saying every integer greater than or equal to $6$ is the sum of three prime numbers. Euler responded by saying an equivalent statement is that even integers greater ...
user avatar
2 votes
0 answers
92 views

Assuming the Goldbach conjectures are true, will all $O$ and $O ⋅ 2$ share at least $1$ of the $p$'s in such a way that the remaining are also $p$'s?

Assuming the Even Goldbach conjecture is true, it will mean that every even number greater than $2$ as $E$ can be represented as the sum of $2$ primes as $p$'s Assuming the Odd Goldbach conjecture is ...
Isaac Brenig's user avatar
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2 votes
0 answers
176 views

Is it possible to prove certain conjectures have no proof?

We will use Goldbach's conjecture as an example$^1$. It is either true or false that every even number greater than 2 is the sum of two primes. Let's take a look at these two scenarios. Goldbach's ...
willmaths's user avatar
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0 answers
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Approach towards proving some symmetries of prime numbers

Reviewing draft documents from years ago, I found an interesting line of reasoning for approaching a proof of what I called the "symmetry conjecture" about prime numbers, which can be stated ...
Juan Moreno's user avatar
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1 vote
1 answer
108 views

A question about proof of Ternary Goldbach Conjecture.

Let's recall: Von Mangoldt function $\Lambda$ is a following function: $$\Lambda: \mathbb{N}\rightarrow \mathbb{R}$$ $$ \Lambda(n) = \left\{ \begin{array}{ll} \log p & \textrm{if $n = p^k$ for ...
mkultra's user avatar
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