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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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### 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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### 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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### 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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### 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{...
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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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### Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes.

Prove that the following conjecture is equivalent to the strong Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes. I'm able to prove it, but i don't have much experience in ...
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### Even numbers sum of two primes [closed]

We don't know if the Goldbach conjecture is true, but do we we know some type of even numbers which can be expressed as sum of two prime numbers (excluding the trivial sums of two prime numbers) ? ...
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### For any $a \in \Bbb{Z}$, can we always find two prime numbers $p, q$, such that $p - q \in (a)$? [duplicate]

This is a major weakening of many prime sum / difference existence questions. Let $a \in \Bbb{Z}$ and $(a)$ the ideal generated by $a$. Then do there exist two primes $p, q$ such that $p - q \in (a)$...
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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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### 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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### Are there any counterexamples to a generalized Goldbach conjecture?

The Goldbach conjecture suggests that every even integer greater than $2$ can be expressed as a sum of two primes. For example: $$10 = 5+5$$ $$12 = 7+5$$ $$14 = 7+7$$ What is so special about even ...
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### Calculating probability of being $m$ and $(n-m)$ both prime in Goldbach conjecture

The Prime Number theorem asserts that an integer $m$ selected at random has roughly a $\frac{1}{\ln{m}}$ chance of being prime.Thus if $n$ is a large even integer and $m$ is a number between $3$ and \$\...