Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set? Goldbach's conjecture said Ev …

What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)? In other words, I am curious about …

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard …

Possible Duplicate: Why is 2 so odd? I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heurist …

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes les …

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is $$ P(k) = \frac{1}{\zeta(k)}. $$ I am looking at a …

Hello Dear there is a conjecture for which I do not know how it is called. The conjecture is: Every even number can be always written as the difference between two prime numbe …

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the su …

Let $p_n$ be the nth prime and $p_L$ be closest to its square root: \begin{equation} p_L^2 \approx p_n \approx x \end{equation} Let $\sigma \in Z^+$ be a positive integer consta …

I would like to know if I could do something like: $\epsilon (0)\text{:=}0$ $\epsilon (n)\text{:=}\frac{4}{2 n-(-1)^n+1}$ and use it instead of a constant. As $n\rightarrow \inf …

I have read that the Riemann Hypothesis is equivalent to $\pi(x)=\text{Li}(x)+O(\sqrt{x}\log x)$ Is there an analogous statement saying the Riemann Hypothesis is equivalent to $ …

Why is the Chebyshev function $\theta(x) = \sum_{p<=x}\log p$ useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking a …

We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where ${p_i}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc.. The first few values of $R_n$ are: 10, 15, 23 …

Irreducible polynomials are often introduced as the analog to prime numbers in polynomial rings. Prime numbers, of course, have a very rich theory, leading to the likes of the Rie …