Intereting Posts

Logic and number theory books
Injectivity is a local property
Proving a scalar function is differentiable at the origin but that its partial derivatives are not continuous at that point.
Tightness and countable intersection of neighborhoods
Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
Difference Between Tensoring and Wedging.
Why are so many of the oldest unsolved problems in mathematics about number theory?
Is it unlikely to get the same number of heads/tails?
A limit problem related to $\log \sec x$
Sobolev Embedding (Case: p=N)
Divisibility question
Differences between infinite-dimensional and finite-dimensional vector spaces
Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
Calculus of Natural Deduction That Works for Empty Structures
Is every set in a separable metric space the union of a perfect set and a set that is at most countable?

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 $n<p<2n-2$.

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.

- What was Ramanujan's solution?
- Does $k=9018009$ have a friend?
- Congruence question with divisibility
- When is the $lcm$ of a fraction sum the actual denominator.
- To prove that $ + ++\dots ++$ is even.
- Are Mersenne prime exponents always odd?

So, is this known?

- How many ways a composite number can be resolved into two factors which are prime to each other?
- if the sum of the digits of $a+b$, $b+c$ and $a+c$ is $<5$, then is the sum of the digits of $a+b+c$ less or equal than 50?
- Smallest multiple whose digits are only ones and zeros
- What is the distribution of leading digits of the squares?
- Computing $n$ such that $\phi(n) = m$
- If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
- Infinite Prime Numbers: With Fermat Numbers
- How prove that $\frac{a_{4n}-1}{a_{2n+1}}$ is integer where $a$ is the Fibonacci sequence
- Why is every answer of $5^k - 2^k$ divisible by 3?
- Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$

```
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 $x \geq 3275.$

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