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 ...
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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.
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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\...
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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 ...
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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 ...
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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 $k$ ...
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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 ...
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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'...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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:
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