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 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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Is the stronger form of Dirichlet's theorem on arithmetic progressions strong enough to prove Goldbach's (asymptotic) conjecture?
The stronger form of Dirichlet's conjecture states that, for example, $$\lim_{N\to\infty} \frac{\text{ the number of primes } \leq N \text{ of the form } 1+8k }{\text{ the number of primes } \leq N \...
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Can this light variant of Goldbach be proven?
Goldbach's conjecture states that every even integer greater than $2$ is the sum of two (not necessarily distinct) prime numbers.
It seems that for $n>6$ , we can choose a representation with ...
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Is the function for the Weak Goldbach Conjecture an increasing function?
Premise
There is a function used to count the number of ways a given odd integer larger than 5 can be written as the sum of three prime numbers.
I have seen the function ($f_{3}$) expressed as the ...
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If an arithmetic function is multiplicative, non-zero at a prime, and "prime-linear", is it the identity?
Let $f:\mathbb{N}\to\mathbb{N}\cup\{0\}$ be a function. Let $f(1)=1,$ and $f(ab)=f(a)f(b)$ whenever $\gcd(a,b)=1.$ Note that I am assuming that $f$ is multiplicative but not completely multiplicative. ...
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Curiosity about the nature of primes going to infinity.
My question is about the nature of primes as you go to infinity.
I was watching a video about the last digits of primes and Chebeshev's bias and I had a thought about the Goldbach conjecture. If N ...
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Why is the function $r(n)$ of particular interest in Circle method?
I was reading Goldbach problem(Ternary version) and encountered the Hardy-Littlewood circle method.In this method,we work with a number $r(n)=\sum\limits_{n=n_1+n_2+n_3}\Lambda(n_1)\Lambda(n_2)\Lambda(...
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Goldbach's conjecture for odd numbers satisfying prime relations
Inspired by this question and remembering Bertrand's postulate, I wondered.
Does any set of random odd numbers satisfying Bertrand's postulate satisfy Goldbach's conjecture (i.e. that every even ...
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Hypothesis to split an even number as the sum of two primes as per Goldbach's Conjecture
I have the following conjecture in regard to Goldbach's Conjecture which I have found via my own experimentation. I wanted to run it by here to see if it is correct and can be proved formally.
$\...
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Smaller Bound to find primes for Goldbach's Conjecture
I am an undergraduate student studying mathematics and have come across an observation. I thought this would be a great place to discuss it.
In my attempt to understand Goldbach's Conjecture in a ...
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A group theoretic formulation for Goldbach's conjecture
Consider the action of multiplying by $(n-1) \pmod n$, in the group of units $U(n)$ when $n$ is an even number. This action is a map $f$ defined on $U(n)$ such that for each $x \in U(n), f(x) = (n-1)x ...
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Goldbach Conjecture-Does the complete graph of odd primes up to 2n with averages as edge labels contain all integers up to $p_{\pi(2n)}$?
Define $G_n=K_{\pi(2n)-1}\circ C_1 $ to be the complete graph on $\pi(2n)-1$ vertices composed with the self-loop to yield a complete graph with self loop edges and where $\pi(k)$ is the prime ...
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Verification of proof below: If $n \in G^1$ and $n >6$, then $\max(A_n) \nmid n$.
Note: I know this is part of Goldbach's Conjecture. Second, I want to prove what the title says.
Define the following sets/functions:
$X = 2\mathbb{Z}^{\ge 4} := \{n \in \mathbb{Z}^+ : n \equiv 0 \...
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Goldbach's Conjecture: Counterexample of "necessary condition"
Definitions:
Divisor Function:
$$\sigma_x(n) = \sum_{d\mid n} d^x$$
Euler's Totient Function:
$$\phi(n) = \# \{m \in \mathbb{Z}^+ \mid (\gcd(m,n)=1) \wedge (1 \le m \le n)\}$$
Conjecture:
The ...
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Proving a variation of Lemoine's Conjecture by assuming the strong Goldbach Conjecture
In 2013, when I was just a totally newbie recreational mathematician, I read about Levy's conjecture (i.e., Lemoine's conjecture, stating that all odd integers greater than 5 can be represented as the ...
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Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.
Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
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An "almost all" result for the binary Goldbach problem
I have a question. My professor in the lecture said that Vinogradov's method by applying the Hardy-Littlewood circle method (minor and major arc) for the ternary Goldbach problem can be used to prove ...
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Equivalences of Goldbach conjecture - Added value?
Let $I(n)$ be the set of the first $n$ odd integers, $S_p(n)$ be a subset of the last $p$ elements of $I(n)$ such that $p$ is some prime number, and $a_{n-\left(\frac{p-1}{2}\right)}$ the $n-\left(\...
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Are there any large numbers found that seemed eerily close to disproving Goldbach's conjecture?
This question is about any numbers that seemed unusually close to disproving Goldbach's conjecture. Meaning any large numbers (say above 100) that had very few sets of primes which satisfied the ...
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Can any positive even number be expressed as an XOR of two prime numbers?
I just came up with this question when I was thinking about the Goldbach conjecture.
For example,
$$2=5 \oplus 7$$
$$4=3 \oplus 7$$
$$6=3 \oplus 5$$
$$8=3 \oplus 11$$
$$10=7 \oplus 13$$
$$12=7 \oplus ...
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Is it true that for sequences that satisfy the property-type in Goldbach's conjecture, there is an integer which cannot be expressed in a unique way?
Let $A\subset \mathbb{Z}$ be such that $\exists\ c\in\mathbb{Z}$ such that $\forall\ n\geq c: \exists\ a,a'\in A$ with $a+a' = n.$ In other words, $A$ satisfies sort-of Goldbach conjecture, but for ...
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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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Why is this infinite series _intuitionistically_ Cauchy?
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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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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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 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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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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