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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.

22
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1answer
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Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
79
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4answers
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Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture

In the most recent numberphile video, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is ...
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0answers
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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 $ord_n(k)$ ...
3
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1answer
282 views

Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?

I am trying to find a counterexample for the following expression when $d=6$. ($G(n)$ = Goldbach partition of the even number $n$) ${\forall}$ n=2*k / k${\in}$N, n${\geq}$8 ${\exists}$(${p_i}$,${p_j}...
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228 views

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} =...
4
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1answer
449 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
3
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0answers
103 views

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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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}^\...
15
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1answer
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Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
12
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7answers
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What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
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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)$...
3
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0answers
190 views

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 $4n+1$ ...
8
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2answers
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Disprove the Twin Prime Conjecture for Exotic Primes

The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form $2^p - 1$ where $p$ is a prime, A000668, $43\%$) Sophie Germain primes ($p$...
6
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1answer
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What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?

All: what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ? In other word, for other algebraic ...
3
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2answers
172 views

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
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0answers
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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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2answers
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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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1answer
125 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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0answers
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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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1answer
228 views

Any results for small number Goldbach conjecture research?

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc). My understanding is that those ...
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1answer
126 views

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 ...
0
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1answer
198 views

Goldbach's Weak Conjecture

I have a few questions on GWC, as the Wikipedia's page on it appears to be somewhat incomplete. Which of the following two statements is considered as the actual GWC? Every odd number greater than 5 ...