# Questions tagged [goldbach-type-problems]

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### Sieve methods and primal height

Define the primal order $\omega_{\mathbb{P}}(n)$ of an integer $n$ as the smallest $k$ such that the $k$-th iterate of the prime counting function $\pi$ evaluated at $n$ is composite and the primal ...
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### Primality radius and Hamming distance

This question is a follow up to About Goldbach's conjecture, whose relevant content I copy paste below so as to make it self-contained: "let's consider a composite natural number $n$ greater or ...
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I'm trying to generalize the notion of primality radius of a positive composite integer $n$, defined as a positive integer $r$ such that both $n-r$ and $n+r$ are prime, to Gaussian integers. As such, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Asymptotics for the $m$-th integer of given fundamental primality radius and order of centrality

Assuming Goldbach's conjecture, define the "fundamental primality radius of $n$", denoted by $r_{0}(n)$, as $\inf\{r>0\mid (n-r,n+r)\in\mathbb{P}^{2}\}$, and the "order of centrality ...
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### How Important was Estermann's Theorem : Almost all even positive integers are sum of two primes and Why?

I am a second year masters student and earlier this year I was reading the research paper : "On Goldbach's Problem : Proof That Almost All Even Positive Integers are Sums of Two Primes" By T....
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### Goldbach conjecture and the representation number

Let $g(2n)$ be the number of representations of $2n=p+q$ with primes $p$ and $q$. Many people have asked whether $g(2n) \ge 2$ when $2n = p+q$ for some primes $p$ and $q$. That is, does $g(2n) \ge 1$ ...
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### Distribution of Goldbach's weak-conjecture's prime-triples

From Harald Helfgott's proof of Goldbach's weak conjecture,1 we know that every odd number $> 7$ is the sum of three odd primes. If $n$ is such an odd number, say that two sums that yield $n$ are ...
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### Primality radii and Sidon sets

I learned tonight what a Sidon set is, in a book about Erdős. This notion inspires me the following question : For $n$ a large enough composite integer, say $r>0$ is a primality radius of $n$ if ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $p$ is a prime that divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive ...
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, ...