# Questions tagged [goldbach-type-problems]

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### A recurrence for the even numbers in terms of a sum of pairs of prime numbers. Is this Goldbach problem related?

Let the first row and the first column in this table below be the characteristic sequence https://oeis.org/A010051 of prime numbers: {0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, ...
349 views

### Goldbach conjecture and the difference of two primes

The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes? That is, if $2n$ is a sum of ...
186 views

### Is the Goldbach conjecture easier if we allow 1 as a prime?

I hope this is the right site for the question. Is the Goldbach conjecture easier if we allow 1 as a prime? (12=1+11 would be allowed as Goldbach sum for 12) IOW: if we can prove Goldbach for the case ...
722 views

### On Buchstab et al's "forgotten" sieve and the Goldbach conjecture for certain integers

There is a somewhat forgotten sieve-theoretic approach to the Goldbach conjecture, due to Buchstab et al, see e.g. pp.247-248 of R.D. James. On p.247, James defines some function $F$ such that for any ...
1 vote
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### Another Goldbach variation for odd numbers?

Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime. Dabbling in the dark art ...
72 views

### Can an upper bound for $r_{0}(n)$ be reached from a duality principle about the distinct primes $n$ "defines"?

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 and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, so that $k_{0}(n)$ ...
71 views

### Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary

Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...
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### Inversion shift of a Galois radius

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ ...
1 vote
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### Arithmeticity of the sequence of primality radii of an integer and upper bound of its lowest term

This question is a follow up to About Goldbach's conjecture. As shown by the numerical computations at the end of that question, $\alpha_{n}\ll_{\varepsilon}n^{1/2+\varepsilon}$ makes the sequence ...
233 views

### Lower bounding the number of Galois radii of an integer

Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$. Does it suffice to ...
669 views

### Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
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### $k$-Taiwan numbers

Say a positive composite integer $n$ is a $k$-Taiwan number if $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}=p^{k}$ for some prime number $p$, and a Taiwan number if it is a $k$-Taiwan number ...
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527 views

### Staircase numbers

Assuming Goldbach's conjecture, denote as usual by $r_{0}(n)$ for any large enough positive integer $n$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime. Let's define the ...
227 views

### A number theoretical identity of exponential sum

I try to understand a number theoretical identity used by Jan-Christoph Schlage-Puchta in this answer. He defined the function $$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$ where $\Lambda(n)$ is ...
105 views

### Does one have $2r_{0}(n)\lesssim k_{0}(n)(\log n)^{1+1/k_{0}(n)}$?

Under Goldbach's conjecture, I'm trying to find an upper bound for $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ that would generalize Cramer's conjecture. Denoting by $k_{0}(n)$ the quantity ...
73 views

### Order of growth of the error term of the log-exponent of the average prime gap

Disclaimer: I might have already asked this question or a very similar one but couldn't find it if it is so. Hope it will be judged somehow interesting anyway. Assuming Goldbach's conjecture, let's ...
1 vote
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1 vote
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### Projection of cocyclic Gaussian primes on the real axis

I just stumbled upon https://math.stackexchange.com/questions/2372062/4-concylic-points-of-gaussian-primes after a quick Google search about cocyclic Gaussian primes. As I've been investigating about ...
310 views

### Prime gap distribution in residue classes and Goldbach-type conjectures

Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question). The general problem that I try to solve is ...
248 views

### A nice pattern about Goldbach conjecture in French Wikipedia

In the following link: https://fr.m.wikipedia.org/wiki/Conjecture_de_Goldbach, one can see a nice pattern of pink and blue lines coming from each prime number, the intersection points thereof are ...
429 views

### Congruential equidistribution, prime numbers, and Goldbach conjecture

Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let $$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$ Here $\chi$...
1 vote
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### Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions: $N_S(z)$ is asymptotic continuous version of the function counting the ...
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Let $S$ be an infinite set of positive integers. Let us define the following quantities: $N_S(z)$ is the number of elements of $S$, less or equal to $z$ $r_S(z)$ if the number of positive integer ...
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Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that $$N_i(S) \sim \frac{a_i z^{... 1 vote 0 answers 75 views ### t-balanced numbers Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture. For n a large enough composite positive integer, write r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}, ... 2 votes 0 answers 142 views ### Is this conjecture equivalent to Polignac's conjecture? Under Goldbach's conjecture denote by r_{0}(n) for n a large enough composite integer the quantity \inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}, by k_{0}(n) the quantity \pi(n+r_{0}(n))-\pi(n-... 0 votes 2 answers 481 views ### Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language I thought of utilizing this lockdown period to study research papers in number theory by myself. I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ... 4 votes 0 answers 140 views ### Can this number be interpreted as a fractal dimension? Under Goldbach's conjecture, let's denote for a large enough integer n by r_{0}(n) the quantity \inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\} and by k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n)). Let's ... -1 votes 1 answer 125 views ### Statement about upper density of even numbers satisfying the Goldbach condition For A\subseteq \mathbb{N}, let the upper density of A be defined by$$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$Let$$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \...
For $n$ a large enough positive composite integer, say $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime. Say $n$ is a karmic number if the following holds: $r$ is a primality radius ...