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### Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....
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### Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...
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Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$\underset{q>P}{\sum}\,\frac{\mu(q)^{2}}{\phi(q)^{2}}\overset{q}{\underset{a=1}{\sum}^{*}}e\... 1answer 180 views ### “Pseudo-random” subsets of additive bases We say that a subset B \subset \mathbb{N} is an (asymptotic) additive basis of order k if the set kB = B + \cdots + B = \mathbb{N} \setminus C, where C is a finite set of positive integers. ... 0answers 134 views ### Which upper bound for r_{0}(n) can be obtained through the Chinese Remainder theorem? Assume Goldbach's conjecture. Then for every integer n greater than one there exists a non negative integer r such that both n-r and n+r are prime numbers. For a given n, let's denote r_{0}(... 2answers 381 views ### Research on the structure of a non-Goldbach number? Has there been any research into the structure of a non-Goldbach number? This seems like it would be a profitable area for proof by contradiction, so I assume that someone has already done it. (i.e. ... 1answer 500 views ### A possible consequence of Dirichlet's theorem about primes in arithmetic progression EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained. "let's consider a composite natural number n greater or equal to 4. ... 0answers 438 views ### A conjecture on the relative size of Goldbach pairs? On leafing through some papers of John Nash (available online on his webpage) I found this intriguing little observation: Noticing that with larger even numbers it seemed to become possible to ... 0answers 332 views ### Divisor function inequality I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let z=N^{1/8}, then$$\sum_{w\leq z}\...
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 ...