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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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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Citizens and rebels: a twin prime related categorization of composites
Assuming Goldbach's conjecture and denoting by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$, consider the sequence $(u_n)_n$ such that $...
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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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119 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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2answers
79 views
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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Lower bound for the ratio of primes and semiprimes as summands
Chen's theorem states that every large enough even integer is the sum of a prime and an almost prime, i.e. an integer that is either a prime or the product of two primes. As there are $s(n):=\pi(2n)-\...
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Is there a counter-example to these number theoretic conjectures?
Question and Summary
I recently made the following heuristic observations:
Let,
$$ xy = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n} $$
where $a_i\geq1$
Conjecture $1$: then there must exist $x-y=p_{n+1}$ ...
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Does the assumption of both Goldbach and Elliott-Halberstam conjectures imply the twin prime conjecture?
Under Goldbach conjecture, denote by $ r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ for any large enough composite integer $ n $ .
Say a positive composite integer $ n $ is hexahedral if ...
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27 views
Goldbach's conjecture and convergence of a Dirichlet series
Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) : =\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\} $. The assumption of GC implies $ r_{0}(n)<n $.
Let's now consider the series $ G(s) : =\...
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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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1answer
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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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How many pairs $(x,y)$ such that $x+y=n$ have an $x$ or $y$ divisible by 3 but $x$ and $y$ are not equal to 3?
Let the set $S_{n}$ = {$(x,y):x,y \in \mathbb{O}$} such that $x+y=n$ where $\mathbb{O}$ is set of odd integers > 1.
Let us define the function $f(n) = |S_{n}|$ that counts the number of pairs in $S_{...
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1answer
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Taking an unproven but “seemingly true” statement as an axiom
I'm sorry for this uneducated question, but I've been thinking of this for a few hours and I couldn't find anything on the topic. Perhaps it is just a failure on my part and a limitation of my ...
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1answer
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Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs?
Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$?
For example,
If we ...
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Thoughts on Lehs conjecture $\forall n>2\in \mathbb N\exists a,b\in \mathbb N$ such that $a+b=n\land (a+ab+b)\in \mathbb P$. Lehs comet?
Lehs conjectured here that $\forall \ n>2\in \mathbb N,\exists\ a,b\in \mathbb N$ such that $a+b=n\land (a+ab+b)\in \mathbb P$. In comments, Crostul restated this as $\forall \ n\ge 4\in \mathbb N,...
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Where is the mistake: On the sum of two prime numbers.
Someone could help me find some error in the reasoning:
We know, that the canonical decomposition of $n!$ is:
$n!=\prod_{p_{i}\leq n}p_{i}^{\alpha_{i}(n)}$, where:
$\alpha_{i}(n)=\sum_{t=1}^{r}[\...
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2answers
205 views
Searching for Goldbug Numbers
A Goldbug Number of order k is an even number 2n for which there exists some k order subset of the prime non-divisors of n $2 < p_1 < p_2 < p_3 < \cdots < p_k < n$ such that $(2n-p_1)...
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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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2answers
326 views
Goldbach's Conjecture and the totient function
A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (...
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Maximal Goldbach Partitions?
A Goldbach partition $2n = p + q$ with $p$ and $q$ primes and $p \leqslant q$ is usually called minimal if the numbers $2n - k$ ($k = 1,\ldots, p-1$) are all composite.
Reading through the literature,...
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1answer
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Is every large enough even integer the sum of two quasiprimes ?
Call a "quasiprime" any integer of the form $ p^{k} $ with $ k\geq 1 $ and $ p $ a prime number. In other words, the set of quasiprimes is exactly the set of integers for which the von Mangoldt $ ...
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1answer
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Time complexity of finding the largest Goldbach partition
Suppose we are given a large even integer $N$, and we want to determine primes $p$ and $q$ such that $N = p + q$, subject to the conditions that $p \geqslant q$ and $p - q$ is as small as possible. (...
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Are those limits defined in terms of a positive integer actually independent thereon?
Under Goldbach's conjecture, denote by $ r_{0}(n)=\inf\{r\gt 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and by $ k_{0}(n)=\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $ for $ n $ large enough. Say $ n $ is $ k $ -...
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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$ ...
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Is there any algorithm to identify the smallest example of gap size $n-1$ between consecutive prime numbers
In contemplating Goldbach's conjecture, I became interested in gaps between successive primes. If $n<a<b<2n$ and the range $a$ to $b$ is a primeless gap, then one could ignore any primes in ...
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Every integer greater than $0$ can be expressed as a sum of $a$'s and $b$'s, if and only if $a$ and $b$ have no common factor
Every integer greater than $0$ can be expressed as a sum of $a$'s and $b$'s, if and only if $a$ and $b$ have no common factor.
PROOF:
Consider the case $a=5$, $b=13$. First, let's find how to ...
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190 views
Why can't we prove Goldbach's conjecture with this method?
1: Prime gap bounds:
Consider the following non-asymptotic bounds for $\pi(x)$, proven by Dusart in 2018 (holding for $x>5393$):
$$\frac{x}{\log(x)-1}<\pi(x)<\frac{x}{\log(x)-1.112}$$
To ...
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1answer
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$O(1)$ algorithm for Goldbach partitions, assuming $\pi(n)$ is known for all $0<n<2n$?
Instead of a list of prime numbers, we will be using $B_{p}(2n)$, a binary representation of the distribution of prime numbers from $3$ to $2n$. Simply writing 0 for non-primes and 1 for primes, we ...
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Lower bound for $g(n)$, the number of decompositions of 2n into ordered sums of two odd primes
I was coding an algorithm that calculates $g(n)$, the number of decomposition of 2n into ordered sums of two odd primes (A002372), or the number of Goldbach partitions. I noticed i can express the ...
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1answer
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Decomposition $2^k$ into a sum of primes
We know Goldbach's_conjecture. It concerns even numbers. However even numbers have many subsets.
My questions concern decomposition of specifically a number $2^k$ into a sum of two primes:
Is $...
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Why Goldbach Conjecture is difficult to solve in $\mathbb{R}[x]$ and $\mathbb{C}[x]$?
In an article on 'Comparing the close cousins $\mathbb{Z}$ and $\mathbb{F}_q[x]$', I've found the following
The fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials ...
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1answer
102 views
Are there any counterexamples to a generalized Goldbach conjecture?
The Goldbach conjecture suggests that every even integer greater than $2$ can be expressed as a sum of two primes. For example:
$$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 $\...
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Goldbach's Conjecture and 1-1 correspondence [closed]
I know math only (somewhat) as a recreation, so I know this is a naive and ignorant question, but I don't have the mathematical terminology or experience to figure out why it has to be incorrect. I am ...
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Algorithm to find prime number of a specific even number in Goldbach's conjecture [closed]
According to Goldbach's conjecture:
Every even integer greater than 2 can be expressed as the sum of two primes.
What is the most efficient algorithm which takes an even number and gives the two ...
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Landau's comment on validity of Goldbach's conjecture
Edmund Landau had once put forth the following statement about the validity of the Goldbach conjecture.
"the goldbach conjecture is false for at most 0 % of all even integers ; this at most 0 % does ...
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An equivalent form of the Goldbach conjecture using the radical of an integer and the Euler's totient function, and a related problem
Let $\phi(n)$ denoting the Euler's totient function and $\operatorname{rad}(n)$ the so-called radical of an integer (see this Wikipedia.)
The MathWorld's article dedicated to the Goldbach conjecture ...
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How to turn number into sum of unique primes?
I have to find algorithm which find prime number less than $n$ which is sum of the largest amount of unique primes, for example for $n=81$, the answer is $79 = 3 + 5 + 7 + 11 + 13 + 17 + 23$.
I have ...
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1answer
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Goldbach's conjecture proof based on a continuous approach and on a discrete approach [closed]
I conducted a study on the Goldbach binary conjecture and now I will show my proposal to demonstrate this conjecture, where P{2} represents the odd prime numbers set. I would like to know if you think ...
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Can you prove or disprove the following list of my conjectures?
The following three statements are my own conjectures, not a homework problem.
$a)$ For $n = 3, 4, 5,..$, every square integer $n^2$ can be expressed as the sum of a prime $p$ and two other primes $...
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Upper bound for the number of representation numbers as the sum of two primes
To prove strong goldbach conjecture one can use a lower bound of number of the representations of a number as the sum of two primes. If its greater than zero, than we have conjecture.
I wonder if ...
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A fun fact relating to Goldbach Conjecture
I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and ...
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250 views
Can we hope for an elementary proof of a conjecture of Goldbach?
It is written on Wikipedia:
"During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, ...
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1answer
59 views
Feedback on a general conjecture for A = Bp – Cq where p and q are primes?
I previously asked for feedback on the following conjectures (similar in appearance to Goldbach ‘s and Lemoine’s see: Two new conjectures related to Lemoine's and Goldbach's) :
$O = 2n + 1 = p ...
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Goldbach Partition - Why Co-Primality?
Any even number $2n$ can be written as the sum of two primes, $p_{a}$ and $p_{b}$. For $n \geq 2$ this is the Goldbach Conjecture.
$$ p_{a} + p_{b} = 2n $$
Why are $p_a$ and $2n$ co-prime? That is, $...
3
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102 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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Almost-norm restatement of Goldbach's conjecture.
Let $p_i$ be the $i$th prime number. Each number $x$ in $\Bbb{Z}$ can be expressed as a finite sum $\sum\limits_i (k_i p_i), \ k_i \in \Bbb{Z}$, in many ways. But define $\|x\| = \min\{ \sum_i |k_i| ...
