Intereting Posts

Limit points in topological space $X$
Rigorous proof that surjectivity implies injectivity for finite sets
Runge's Theorem for meromrophic functions
What are the practical applications of this trigonometric identity?
calculating the probability of k changeovers when flipping a coin
Integral $\int_0^1 \log x \frac{(1+x^2)x^{c-2}}{1-x^{2c}}dx=-\left(\frac{\pi}{2c}\right)^2\sec ^2 \frac{\pi}{2c}$
Number of reflection symmetries of a basketball
52-card deck probability…
A Question on Compact Operators
Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$
how to visualize binomial theorem geometrically?
Find all integers satisfying $m^2=n_1^2+n_1n_2+n_2^2$
Euler's Phi Function Worst Case
bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$
Categorification of the (co-)induced topology

Is it known is there always a prime strictly between $p_n$ and $p_n+n$, where $p_n$ is the $n$-th prime number and $n\geq5$?

I know about Bertrand`s postulate which states that for any integer $n>3$ there is always a prime $p$ such that $n<p<2n-2$.

- Is there any result, that says that $\lfloor e^{n} \rfloor$ is never a prime for $n>2$?
- $p$-Splittable Integers
- Disprove the Twin Prime Conjecture for Exotic Primes
- Primes for >1 (good expression)
- Eigenvalues appear when the dimension of the Prime Index Matrix is a prime-th prime. Why?
- Density of products of a certain set of primes

If we would plug $p_n$ instead of $n$ we would get $p_n<p<2p_n-2$ but since I guess $p_n+n<2p_n-2$ will hold for all but finitely many $n$ we have that this problem of mine is stronger than Bertrand`s postulate and it seems that it is not implied by it.

So, is this known?

- Divisibility of binomial coefficient by prime power - Kummer's theorem
- Prove that $n^2+n+41$ is prime for $n<40$
- Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime
- Infiniteness of non-twin primes.
- Are there infinitely many Mersenne primes?
- If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$
- Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?
- Prove the converse of Wilson's Theorem

```
jagy@phobeusjunior:~$
p p + n n
2 3 1 p/log p 2.885390081777927 p/log^2 p 4.162737962011215
NONONONONO 3
3 5 2 p/log p 2.730717679880512 p/log^2 p 2.485606349070669
NONONONONO 5
5 8 3 p/log p 3.106674672798059 p/log^2 p 1.930285504520985
7 11 4 p/log p 3.597288396588255 p/log^2 p 1.848640544032643
NONONONONO 11
11 16 5 p/log p 4.587356305666709 p/log^2 p 1.913076170467284
13 19 6 p/log p 5.06832618826664 p/log^2 p 1.975994642359189
17 24 7 p/log p 6.00025410570094 p/log^2 p 2.117826431351823
19 27 8 p/log p 6.452842166007064 p/log^2 p 2.191535369442038
23 32 9 p/log p 7.335366744787423 p/log^2 p 2.339461099153619
29 39 10 p/log p 8.612251926827733 p/log^2 p 2.557616663832689
31 42 11 p/log p 9.027406962818835 p/log^2 p 2.628841176527419
37 49 12 p/log p 10.24670205612772 p/log^2 p 2.837700081812219
41 54 13 p/log p 11.04058283064003 p/log^2 p 2.973035835127401
43 57 14 p/log p 11.43252118401864 p/log^2 p 3.039593967977556
47 62 15 p/log p 12.20732420209676 p/log^2 p 3.170612003725475
53 69 16 p/log p 13.34914438384008 p/log^2 p 3.362257656237909
59 76 17 p/log p 14.46951764677999 p/log^2 p 3.548592219160638
61 79 18 p/log p 14.83869415980414 p/log^2 p 3.609620399478779
67 86 19 p/log p 15.93457740311697 p/log^2 p 3.789712791282478
71 91 20 p/log p 16.65618860623435 p/log^2 p 3.907445336428887
73 94 21 p/log p 17.01449483472118 p/log^2 p 3.965658006585665
79 101 22 p/log p 18.08008761459324 p/log^2 p 4.137842634827439
83 106 23 p/log p 18.7832074896648 p/log^2 p 4.250709440961442
89 113 24 p/log p 19.82784807433869 p/log^2 p 4.417343362461308
97 122 25 p/log p 21.20352530592732 p/log^2 p 4.634943148444331
p p + n n
```

From Dusart’s results, we need check only for $p < 4000.$ In that range, we always get $p_{n+1} \leq p_n + n,$ with equality only at $p_{n+1} = 3,5,11.$

As you can see in the output, $n$ is about $p / \log p$ and much larger than $p / \log^2 p,$ even for fairly small numbers. Indeed, from Rosser and Schoenfeld (1962) we have, for $n \geq 6,$

$$ p_n > n \log n, $$ but

$$ p_n < n \log n + n \log \log n $$

yes. First, this is reasonable, as $n \approx \frac{p}{\log p}.$ That is, you are asking, more or less, whether we have a prime between $p$ and $p + \frac{p}{\log p} $

We have the result of Dusart, page 8, theorem 6.8, that there is a prime between $x$ and

$$ x + \frac{x}{25 \log^2 x} $$

as long as $x \geq 396738.$

Here we go, Pierre Dusart in his Ph. D. dissertation, for a lower bound gives a milder outcome that is good enough,

there is prime between $x$ and

$$ x + \frac{x}{2 \log^2 x} $$

as long as $x \geq 3275.$

BETTER BERTRAND

- Positive definite matrix must be Hermitian
- Measure on topological spaces
- The number of real roots of $1+x/1!+x^2/2!+x^3/3! + \cdots + x^6/6! =0$
- Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$
- Can you recommend some books on elliptic function?
- If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$
- Question About Concave Functions
- Slick proof the determinant is an irreducible polynomial
- How to prove that inverse Fourier transform of “1” is delta funstion?
- What is the Arg of $\sqrt{(t-1)(t-2)}$ at the point $t=0$?
- Help me solve a combinatorial problem
- Intuitive explanation of the Fundamental Theorem of Linear Algebra
- What's the difference between predicate and propositional logic?
- Showing that a homogenous ideal is prime.
- Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.