# Tagged Questions

Prime numbers are natural numbers not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

103 views

### Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
24 views

### If all elements (barring identity) are of equal prime order then $|G|=p^k$

For a finite group $G$, if $|g|=p$ for all $g\in G$ except for $e$ , where $p$ is a prime, then $|G|=p^k$ for some integer $k$. I've been thinking how this can be proved, but so far with not much ...
19 views

### $\lfloor\log{p_{n}}\rfloor$ having more than one solution for individual $k$

Question: If you assume that a. $k\in\Bbb{N}$ b. $p_n$ denotes the $n$'th prime number. $p_0$ doesn't exist. c. $n\in\Bbb{N}$ I am fairly certain that: At least two distinct integer values for ...
205 views

### Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
35 views

25 views

### $\langle ab + 1 : a,b \text{ prime}\rangle$ is not a finitely generated subsemigroup of $\Bbb{Z}^{\times}$.

Let $T \equiv PP + 1 \equiv \{ ab + 1 : a,b \text{ are prime }\} \subset \Bbb{Z}^{\times}$. Consider the subsemigroup generated by $T$. How can I show that it is not finitely generated, by that I ...
### On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number
In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...