# 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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"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 \... 0 votes 0 answers 53 views ### 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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Disclaimer: this was first asked on MO and got downvoted there. This question deals with a weakening of the notion of primality radius of a composite integer $n$, defined as a positive integer $r$ ...
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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 ...
1 vote
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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, ...
1 vote
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### Does this upper limit equal $1$?

Under Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ where $\pi$ is the ...
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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 ...
1 vote
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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 ... 72 views

### 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 ...
1 vote
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### A motivating argument for studying determinants of $2\times 2$ matrices with prime entries in relation to Goldbach's conjecture.

Let $\Bbb{P}$ be the set of odd primes. Let $X_n$ for $n \geq 3$ be the Goldbach solution set $X_n = \{(p,q) \in \Bbb{P}\times \Bbb{P} : 2n = p + q \}$. Suppose that for combinatorial reasons we are ...
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### About Cohen's proof for Goldbach's conjecture

As a context, my advisor recently lent me the book Uncle Petros & Goldbach's Conjecture to read, a story about a man obsessed trying to prove/disprove the Goldbach conjecture. I was searching ...
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### Goldbach conjecture and other problems in additive combinatorics

The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance: $S = T$ is the set ...
1 vote
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### Statement regarding Goldbach's Conjecture?

Question I think using elementary (but twisted) means I can prove an interesting statement and was curious how a number theorist would prove the same. Let us we want to find $2$ primes which ...
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### Conjecture: All but 21 non-square integers are the sum of a square and a prime

Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here. I came up with the following ...
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### Can any number of the form $2p$, for $p>3$ a prime, be written as the sum of two distinct primes? [duplicate]

I think Goldbach's conjecture is quite well-know at this point, but there is no problem restating it: any even integer greater than $2$ can be written as the sum of two prime numbers. But what about ...
1 vote
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### For what percentage of numbers does this proof of Goldbach's conjecture hold?

Question For what percentage of numbers does the below inequality hold? $$\pi(2m) > \frac{\phi(2 m) -1}{2}$$ where $m$ is not a prime or $1$, $\pi(m)$ is the number of primes less than $m$ and ...
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### Can a number N be expressed as sum of different primes. And if so, can we say goldbach conjecture is just a special case of it?

I was looking into Goldbach Conjecture proof. Is it proved that we can express any integer as a sum of different primes. And if yes, Goldbach conjecture says we can do that with even numbers and ...
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### Relationship between Goldbach conjecture and Mersenne primes

I was reading about Mersenne primes and wondering about its connection with the Goldbach conjecture. It's conjectured that there's infinitely many Mersenne primes. One direction is trivial, that ...
1 vote
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