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 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,\...
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Proof by exhaustion under Intuitionism

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 ...
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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 ...
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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 ...
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6 votes
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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(...
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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}...
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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 \...
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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,...
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Weak primality radius

Disclaimer: this was first asked on MO and got downvoted there. This question deals with a weakening of the notion of primality radius of a composite integer $n$, defined as a positive integer $r$ ...
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Why doesn't proof of the weak Goldbach conjecture also prove the strong Goldbach conjecture?

The weak Goldbach conjecture can take the representation O = a + b + c, where O is an odd number greater than 5, and a, b, c are prime. It can thus also take the form O - a = b + c. Since at least one ...
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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, ...
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Does this upper limit equal $1$?

Under Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ where $\pi$ is the ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Can any odd positive integer greater than $3$ be expressed as the sum of $2$ perfect square numbers (excluding $0$) plus $1$ prime number?

I have been reading about the "odd Goldbach conjecture" which states that: Every odd integer greater than $7$ can be written as the sum of three odd primes I have also been reading about the ...
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Has it been proven that all positive even numbers > 2 are either the sum of two primes or the difference between two primes?

If a Maillet number is a positive even integer that can be expressed as the difference between two primes and a Goldbach number is a positive even integer that can be expressed as the sum of two ...
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prime divisors of the sum of two integers with given prime divisors

I want some results for the following statement. Let $S_{1}$, $S_{2}$, and $S_{3}$ be set of primes. There are infinitely many pairs of two integers $\left(m,n\right)$ such that (1) any prime divisor ...
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What is the limit of this Goldbach Conjecture related Dirichlet series?

Background Let us recollect the Goldbach's conjecture. For any $n \geq 2$ there exists primes $p_i$ and $p_j$ such that: $$ 2n = p_i + p_j $$ where $p_k$ is the $k$'th prime. Now we define a function $...
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Are these Goldbach conjecture based quantities related to Euler's constant?

Under Goldbach's conjecture, let's denote by $r_{0}(n)$ for any large enough composite integer the smallest positive integer $r$ such that both $n+r$ and $n-r$ are prime, and by $k_{0}(n)$ the ...
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2 votes
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For a given positive integer $n$, do there exist positive integers $x,y$ st $\phi(x)+\phi(y)=2n$?

The question is: for a given positive integer $n$, do there always exist positive integers $x,y$ such that $\phi(x)+\phi(y)=2n$? The case $n=1$ is good using $x=y=1$. Assume $n>1$. If we invoke ...
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The number of ways an even integer can be written as the sum of two primes

We all know the famous Goldbach Conjecture, namely that every even intger $>2$ is the sum of two primes. I was recently playing around with this conjecture and found out that there are MANY ways of ...
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Why is this proof of Goldbach's Conjecture flawed?

I know just as much as the next guy that this will be a far way off the mark, which is why I have phrased the question as "why is this wrong", not "is this wrong" - so let's do ...
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Decider that reject All inputs if and only if Goldbach's conjecture has a counter example

I want to design a $Decider$ such that for every input accept if and only if Goldbach's conjecture is true and reject all inputs if and only if Goldbach's conjecture has a counter example. I made a $...
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Question on strong Goldbach conjecture

Lately I realized that strong Goldbach conjecture could be "reduced" to show that every even composite number with more than two prime factors (not necessarily distinct) can be expressed as ...
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Is there a conjecture or result such that if it's proven to be true would automatically prove (the Strong) Goldbach's conjecture? [duplicate]

I'm just wondering if there's any existing conjectures or results such that if they're proven to be true, Goldbach's conjecture would also be proven to be true. Additional Information: I've looked at ...
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Does this give an approximate value of $r_{0}(n)$?

Let $n$ be a large enough composite integer and assume Goldbach's conjecture. Denote the smallest $r>0$ such that both $n-r$ and $n+r$ are prime by $r_{0}(n)$, and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(...
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1 vote
1 answer
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Why it is important to prove Goldbach conjecture?

There are a number of questions here, but they are all related. Why it is important to prove the Goldbach conjecture? One reason, I came across is - It will help determine distribution of prime ...
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About stronger version of Goldbach conjecture

I was recently looking at a elementary number theory book and there was a conjecture which amazed me. That conjecture is nothing but Goldbach's conjecture which says that Every even number greater ...
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Goldbach's conjecture and how to express it formally

Goldbach's conjecture states that $$\text{Every even integer greater than 2 is the sum of two primes}$$ Is it true to say that Goldbach's conjecture is formally claiming that: $$\forall 2(k+1),\exists\...
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A motivating argument for studying determinants of $2\times 2$ matrices with prime entries in relation to Goldbach's conjecture.

Let $\Bbb{P}$ be the set of odd primes. Let $X_n$ for $n \geq 3$ be the Goldbach solution set $X_n = \{(p,q) \in \Bbb{P}\times \Bbb{P} : 2n = p + q \}$. Suppose that for combinatorial reasons we are ...
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About Cohen's proof for Goldbach's conjecture

As a context, my advisor recently lent me the book Uncle Petros & Goldbach's Conjecture to read, a story about a man obsessed trying to prove/disprove the Goldbach conjecture. I was searching ...
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2 votes
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Goldbach conjecture and other problems in additive combinatorics

The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance: $S = T$ is the set ...
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Statement regarding Goldbach's Conjecture?

Question I think using elementary (but twisted) means I can prove an interesting statement and was curious how a number theorist would prove the same. Let us we want to find $2$ primes which ...
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8 votes
2 answers
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Conjecture: All but 21 non-square integers are the sum of a square and a prime

Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here. I came up with the following ...
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Can any number of the form $2p$, for $p>3$ a prime, be written as the sum of two distinct primes? [duplicate]

I think Goldbach's conjecture is quite well-know at this point, but there is no problem restating it: any even integer greater than $2$ can be written as the sum of two prime numbers. But what about ...
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For what percentage of numbers does this proof of Goldbach's conjecture hold?

Question For what percentage of numbers does the below inequality hold? $$ \pi(2m) > \frac{\phi(2 m) -1}{2} $$ where $m$ is not a prime or $1$, $\pi(m)$ is the number of primes less than $m$ and ...
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Can a number N be expressed as sum of different primes. And if so, can we say goldbach conjecture is just a special case of it?

I was looking into Goldbach Conjecture proof. Is it proved that we can express any integer as a sum of different primes. And if yes, Goldbach conjecture says we can do that with even numbers and ...
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Relationship between Goldbach conjecture and Mersenne primes

I was reading about Mersenne primes and wondering about its connection with the Goldbach conjecture. It's conjectured that there's infinitely many Mersenne primes. One direction is trivial, that ...
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1 answer
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Help needed in deducing an inequality in a research paper in additive number theory

I need help in proving this inequality-> Assuming q to be any positive integer prove that ( 1+ log(q) ) d(q) $\phi^{-2} (q) $ $\leq$ $ q^{-5/3}$ . where d(q) means number of divsor of q and $\phi(...
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Implications of the equivalence of Goldbach's conjecture and an assertion of the zero product property.

Using suitably defined matrices $A$ and $B$, Goldbach's conjecture can be written as: $$(AB)^2 = 0 \to (AB)=0$$ Considering a matrix $C=AB$, this can be written as: $$ C^2 = 0 \to C = 0$$ which is ...
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What results from sieve theory did Chen actually use in the proof of his theorem?

Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). On that page, it is also ...
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Scale exponent conjecture

Under Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ and $n_{r}:=\inf\{n>2,r_{0}(n)=r\}$. Let $e_{0}(n):=\left(\...
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2 answers
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Check that Goldbach Conjecture holds for every even number between 900 and 1000 [closed]

Like in the title, I have to check that Goldbach Conjecture holds for every even number between 900 and 1000. It's worth to note that I'm not supposed to use computer, so writing e.g. Python code ...
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Goldbach's conjecture and Euler's Totient function

A few weeks ago, I found this question in math.stackexchange: Goldbach Conjecture: Subsets of the Euler Totient Function I can not find the relationship between the question asked and Goldbach's ...
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