# Tagged Questions

The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.

3k views

### Why do primes dislike dividing the sum of all the preceding primes?

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 less than $10^9$, I ...
3k views

298 views

302 views

### Estimate on the prime-counting function $\psi(x)$.

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) and if so, where I ...
227 views

### Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...
335 views

### How many primes have the form $(2^p+1)/3$?

Assuming that $p$ is an odd prime. How many primes have the form $(2^p+1)/3$? Is the number finite? Mathematica calculation shows that there are 23 such primes when $p$ ranges over the first 500 ...
440 views

### Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
178 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
142 views

### Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2)$?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function. Define $$t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right|$$ Is it consistent with current ...
509 views

### Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set?

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 Every even integer ...
220 views

### How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional ‎constraint‎?

In part of my research, I need to count (or find a polynomial bound for) the number of ‎possible ‎ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which ‎are ...