Questions tagged [goldbach-type-problems]
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-2
votes
1answer
56 views
Approximation for $ \inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ by minimizing a distance
Under Goldbach's conjecture, let $r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and $k_{0}(n) : =\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $. The PNT implies that one can expect to have $ \dfrac{...
2
votes
2answers
695 views
If Goldbach's Conjecture is eventually true, is it necessarily true?
We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite ...
0
votes
1answer
121 views
Sergei numbers : even integers n being a prime gap at least n times
Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...
-3
votes
1answer
177 views
Can this weakening of Polignac's conjecture be proven?
Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
2
votes
0answers
107 views
Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?
In this previous question of mine
I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture ...
0
votes
1answer
165 views
Upper bound for $\sum r_{0}(n)$
Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound ...
4
votes
0answers
225 views
Error term for Vinogradov's three prime theorem
It can be shown that $$\sum_{a + b + c = N}\Lambda(a)\Lambda(b)\Lambda(c) = \frac{1}{2}\mathfrak{S}(N)N^2 + O(N^2\log^{-A} N)$$ for some $1\ll \mathfrak S(N)\ll 1$using the circle method. Are there ...
11
votes
0answers
257 views
On a relaxed form of Goldbach's conjecture proposed by Erdős
The Goldbach's conjecture says that:
"Every even integer greater that $2$ is the sum of two prime numbers".
Let $\varphi$ denote the Euler's totient function. I remember that a long time ago I read ...
6
votes
0answers
250 views
Is every powerful number the sum of a powerful number and a prime?
A positive integer $n$ is called powerful (OEIS: A001694) if $p^2$ divides $n$ whenever $p$ divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive integers. By ...
4
votes
1answer
525 views
The prime gap 2 and the prime gap 4, are they equally common?
This question was posed today by a student of mine, and I have not been able to find any relevant references.
Let $p_1, p_2, p_3, \ldots$ be the sequence of prime numbers listed in increasing order, ...
2
votes
1answer
107 views
Upper bound for an exponential sum in Waring-Goldbach problem
In Waring's problem, we have Hua's estimate
$$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$
where $(a,q)=1$.
?Do you know a similar upper bound for the sum
$$...
0
votes
1answer
244 views
Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture
Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that ...
6
votes
1answer
324 views
How small can the smallest of the three “weak Goldbach” primes always be?
I've checked here for discussions of Helfgott's proof of the weak GC and found nothing that helps me with the following; apologies if I missed something.
I'm probably being naive here (please ...
0
votes
1answer
366 views
Asymptotics of “ugly” function elucidate Goldbach's conjecture?
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...
1
vote
3answers
679 views
Which even numbers are known to be both prime gaps and the sum of 2 primes?
Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
4
votes
1answer
291 views
Goldbach for certain classes of $n$
Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, Hugh ...
0
votes
0answers
130 views
Expliciting the distance between consecutive Goldbach numbers assuming it's finite
In this paper, the author shows unconditionally that at least one of the following statements holds:
i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...
2
votes
1answer
198 views
Equivalence of Polignac to finite Goldbach?
Is Polignac's conjecture equivalent to a finite form of Goldbach? There is some discussion here as to the difference between Polignac & general Goldbach, but the similarity seems particularly ...
4
votes
0answers
539 views
Is there a hidden symmetry in the prime numbers distribution?
Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...
9
votes
2answers
788 views
Is $n = p-q$ equivalent to Goldbach's Conjecture?
One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than ...
0
votes
0answers
206 views
Does $\pi(n+r)+\pi(n-r)$ decrease as $r$ increases?
Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a ...
7
votes
1answer
651 views
Is this weak asymptotic Goldbach's conjecture open?
Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...
0
votes
2answers
268 views
Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$
Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
0
votes
1answer
353 views
Is a certain sumset derived from primes of a certain form the set of all naturals?
OEIS A167055 Numbers n such that $12n + 5$ is prime.
$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$.
I conjecture that the set of the sum of every two items of this ...
2
votes
1answer
487 views
Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture?
The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in https://mathoverflow.net/questions/163211/upper-bound-for-r-...
8
votes
1answer
951 views
What keeps asymptotic Goldbach's conjecture out of reach of current technology?
Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
3
votes
0answers
267 views
An inequality about Goldbach conjecture
Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq N}{...
1
vote
1answer
193 views
Tail of singular series of Goldbach problem
Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ \underset{q>P}{\sum}\,\frac{\mu(q)^{2}}{\phi(q)^{2}}\overset{q}{\underset{a=1}{\sum}^{*}}e\...
4
votes
0answers
210 views
“Pseudo-random” subsets of additive bases
We say that a subset $B \subset \mathbb{N}$ is an (asymptotic) additive basis of order $k$ if the set $kB = B + \cdots + B = \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. ...
1
vote
0answers
185 views
Which upper bound for $r_{0}(n)$ can be obtained through the Chinese Remainder theorem?
Assume Goldbach's conjecture. Then for every integer $n$ greater than one there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are prime numbers. For a given $n$, let's denote $r_{0}(...
2
votes
2answers
397 views
Research on the structure of a non-Goldbach number?
Has there been any research into the structure of a non-Goldbach number? This seems like it would be a profitable area for proof by contradiction, so I assume that someone has already done it. (i.e. ...
0
votes
1answer
577 views
A possible consequence of Dirichlet's theorem about primes in arithmetic progression
EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained.
"let's consider a composite natural number $n$ greater or equal to $4$. ...
5
votes
0answers
477 views
A conjecture on the relative size of Goldbach pairs?
On leafing through some papers of John Nash (available online on his webpage) I found this intriguing little observation:
Noticing that with larger even numbers it seemed to become
possible to ...
0
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0answers
390 views
Divisor function inequality
I have been reading a paper on the Goldbach conjecture found at
http://people.exeter.ac.uk/pt224/Goldbach.pdf.
At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then
$$\sum_{w\leq z}\...
4
votes
1answer
514 views
The minimal Goldbach basis
Let $n \in \mathbb{N}, n \geq 2$. By minimal Goldbach basis $G_{2n}$(if it is nonempty) of $2n$ , I mean the minimal set of primes such that every even number less than or equal to $2n$ can be written ...
