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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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Goldbach Conjecture related modular arithmetic problem

The modular arithmetic problem is related to the construction of a certain integer $\delta$ for some given $n$ such that $n \pm \delta$ are both positive primes. The problem is: We will say $X \equiv ...
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89 views

Goldbach's conjecture if 1 is counted as prime

I was grading some homework from a Survey of Mathematics course. They were asked to verify that Goldbach's conjecture holds for the first 15 even numbers greater than or equal to 4. A couple of ...
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129 views

Why problem with simple formulation is so hard?

If you ever heard about Collatz conjecture, you know that it is understandable even for middle school students, but no one has solved it yet. The problem is to prove or to disprove that starting with ...
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62 views

Extension of Goldbach's conjecture to polynomials

I noticed that a slightly modified version of Goldbach's conjecture seems to hold for the quadratic $x^2+1$. Specifically, I assert for any even $n\geq 4$, there exists at least one pair $p,q\in\...
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1answer
145 views

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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126 views

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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51 views

Approximating $r_{0}(n)$ with an integral

I'm still trying to find a tight upper bound for the quantity $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. My idea is that one should have $\sum_{r=1}^{r_{0}(n)}\Lambda(n-r)\Lambda(n+r)\...
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34 views

How to prove that $r_{0}(n)/n<1/2$?

Under Goldbach's conjecture, define for a large enough integer $n>n_0$ the quantity $r_{0}(n)$ as $\inf\{r\geqslant 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. Is it possible to prove that $\forall n>n_{...
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1answer
50 views

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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1answer
126 views

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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1answer
128 views

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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116 views

Can it be shown that Double Goldbug Numbers cannot exist?

Define a Double 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 < ...
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1answer
132 views

Goldbach's conjecture among primes only

We couldn't so far prove that every even integer greater than 2 is the sum of two primes. Can something more be said about the following weaker form? Let $r,p,q$ denote primes $$\forall r\exists p,q\...
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On the Goldbach conjecture

As per wikipedia, Goldbach conjecture has been shown to hold for all integers less than $4\times 10^{18}$ since 2012. Has it been verified for some larger bounded integers ?
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Question on the ternary Goldbach conjecture

Is it true that each odd number $m>5$ is sum or three primes $p_i+p_j+p_k$ one of which is always $2$? And why?
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81 views

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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54 views

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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1answer
175 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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124 views

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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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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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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1answer
163 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 ...
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331 views

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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1answer
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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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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
81 views

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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300 views

What is the best algorithm for finding 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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1answer
72 views

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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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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57 views

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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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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1answer
82 views

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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221 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
96 views

$O(1)$ 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
159 views

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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59 views

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
130 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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1answer
92 views

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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3answers
50 views

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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233 views

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 ...