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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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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How many lunar numbers are there up to $x$?

Under Goldbach's conjecture, denote, for any large enough composite integer $n$, by $r_{0}(n)$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime and by $r_{i+1}(n)$ the ...
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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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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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1answer
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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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Congruential equidistribution in infinite sets or sequences of positive integers

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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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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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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64 views

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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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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Another Way to Phrase the Goldbach Conjecture? [duplicate]

The conjecture says that all even numbers are the sum of 2 primes. But is it equivalent to say that a prime number exists equidistantly from every natural number greater than 1, where the distance is ...
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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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106 views

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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51 views

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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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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100 views

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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A Goldbach-like conjecture for Ramanujan primes

Few days ago I wondered if it is known a variant of Goldbach's conjecture but just using Ramanujan primes (in the spirit of the following conjecture). The related articles from Wikipedia are Goldbach'...
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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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1answer
47 views

Sum over n below x of the sum of reciprocals of primality radii of n

Under Goldbach's conjecture, let's denote for $n$ a positive integer greater than $1$ by $\mathbb{G}(n)$ the set of positive integers $r$ such that both $n-r$ and $n+r$ are prime. Denoting by $S(x):=\...
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227 views

Need help with this Goldbach approach.

I have plotted the counts of Goldbach $2n$ sums of primes, below. It is easy to see that the counts of composites gradually decreases relative to $n$ and would like a suggestion on how to prove ...
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Defining the multiplicative identity of $2k$ in terms of its prime summands $a,b$ for Goldbach's conjecture

Making use of the fact $\phi(p) = p-1$, might we not substitute the multiplicative identity of a positive integer, that being 1, with a function which takes into account the prime summands of each ...
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Heuristic argument on Goldbach's conjecture [closed]

If $p$ is prime number $p \not = 2,3$ then $p= \pm 1 \bmod 6$ If $n$ is even: case I $n= 0 \bmod 6$ $n=(+1+6 \cdot a)+(-1+6 \cdot b)=p_a+p_b$ $p_a= 1 \bmod 6,p_b= -1 \bmod 6$ case II $n= 2 \...
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Is it true that $p+q=2n$ for $p<q$ and $p,q,n\in\mathbb P$?

Title pretty much says it. Do there exist primes $p,q$ such that $p+q=2n$, where $p<q$ and $n\geq 4$ and $p,q,n\in\mathbb P$? This is obviously a special case of Goldbach. I'm wondering whether ...
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A question regarding Goldbach's conjecture and an equivalent way of writing it

I was thinking about how a even number, $2n$, can be represented as the sum of two natural numbers in $n$ ways, and that $\forall a \in \Bbb N \Rightarrow \exists b \in \Bbb N:a+b=2n$, where $a,b<...
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Asymptotic bounds for $g(n)$, the number of Goldbach partitions of even integer $2n$.

I explained in another question how I got to this formula for $g(n)$, the number of Goldbach partitions of even integer $2n$: $$g_{\left(n\right)}=\sum_{p\leq2n-3}\omega\left(2n-p\right)-\sum_{p\...
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Need help interpreting this formula for the number of Goldbach partitions

1: Formula for the number of Goldbach partitions. Let $g\left(n\right)$ denote the number of Goldbach partitions of even integer $2n$: $$g_{\left(n\right)}=\sum_{3\leq p\leq2n-3}\left[\pi\left(2n-p\...
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Goldbach Conjecture related modular arithmetic problem

The modular arithmetic problem is related to the construction of a certain integer $\delta$ for some given $n$ such that $n \pm \delta$ are both positive primes. The problem is: We will say $X \equiv ...
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Goldbach's conjecture if 1 is counted as prime

I was grading some homework from a Survey of Mathematics course. They were asked to verify that Goldbach's conjecture holds for the first 15 even numbers greater than or equal to 4. A couple of ...
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Why problem with simple formulation is so hard?

If you ever heard about Collatz conjecture, you know that it is understandable even for middle school students, but no one has solved it yet. The problem is to prove or to disprove that starting with ...
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Extension of Goldbach's conjecture to polynomials

I noticed that a slightly modified version of Goldbach's conjecture seems to hold for the quadratic $x^2+1$. Specifically, I assert for any even $n\geq 4$, there exists at least one pair $p,q\in\...
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Has Goldbach's Conjecture been proven?

When I searched for the proofs for Goldbach's Conjecture, there seems to be a handful (or more) of papers that attempt to solve it. Are there any official proofs out there yet?
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Every sufficiently large positive integer is the average of $n$ distinct primes for certain $n \geq 2$?

I want to generalize a stronger Goldbach's conjecture a little bit because that might help solve it. I was thinking: For all $n \geq 2$, every sufficiently large positive integer $x \geq b_n$ is ...
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Approximating $r_{0}(n)$ with an integral

I'm still trying to find a tight upper bound for the quantity $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. My idea is that one should have $\sum_{r=1}^{r_{0}(n)}\Lambda(n-r)\Lambda(n+r)\...
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Probability that $~r_{0}(n)>n/k$

For a positive integer greater than $1$, let, under Goldbach's conjecture, $r_{0}(n):=\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\}$. What is the probability $P_{k}(n)$ that $r_{0}(n)>n/k$ where $k$ ...
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On Number Theory and Goldbach's Conjecture [duplicate]

Good day. My question is: Does every even number have the form $n=p+ 2+q$, or $n=p- 2+q$ with $p, q $ prime numbers, such that $p\pm2$ is prime number? Up to $n = 1000$ I know that it is true, and it ...
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Probability that $2n$ has no Goldbach partitions.

I'm trying to evaluate the probability that some even integer $2n$ has no Goldbach partitions using the following approach... First, visualize the distribution of primes from $1$ to $n$ as a binary ...
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150 views

Goldbach's conjecture among primes only

We couldn't so far prove that every even integer greater than 2 is the sum of two primes. Can something more be said about the following weaker form? Let $r,p,q$ denote primes $$\forall r\exists p,q\...
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On the Goldbach conjecture

As per wikipedia, Goldbach conjecture has been shown to hold for all integers less than $4\times 10^{18}$ since 2012. Has it been verified for some larger bounded integers ?
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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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Prove the existence of two prime numbers whose product is less that a given integer s.t. the following quantity is a perfect square

I am working on understanding Goldbach's conjecture and trying to make a small project on its various properties. Finally, I came up with the following statement, "Let, $n>2$ be any natural number....
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187 views

Can the solution to $n^2=pq+y^2$ help with the Golbach conjecture?

This question was inspired by the following question. https://mathoverflow.net/questions/132532/goldbachs-conjecture-and-eulers-idoneal-numbers Here, we are not looking to factor an integer $N$. ...
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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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Reaching higher even numbers in Goldbach's conjecture, using lower even numbers.

Let $n \in \Bbb{N}, n \gt 1$. Let $\Bbb{P} = $ the prime numbers in $\Bbb{N}$. Define \begin{align*} A_n &= \{ (p,q) \in \Bbb{P}^2 : p + q = 2n\}, \\ B_n &= \{ (p, q) : p - q = 2n \}. \end{...