Generalizing conjecture of Bertrand

Question

Joseph Bertrand's conjecture for primes from 1845, $\;p_{n+1} < 2p_n$, proved by Chebyshev 1852, can be generalized as follows: $$\forall a\in\Bbb Z_+\exists N\in\Bbb Z_+:(n\ge N\implies\exists p\in\Bbb P: ap_n<p<(a+1)p_n)$$ It seems to be proved for $a=2,3$ that $N=1$ works (Wikipedia).

Computer tests for $p_n< 10,000,000$ $(n\le 664,579)$:
$a=5,9,14$ holds for all primes $p_n\ge 2$, that is, it seems to hold for $N=1$.
The case $a=4$ is tested for $p_n\ge 5\; (N=3)$. The case $a=25$ is tested for $p_n\ge 23\; (N=9)$.

Computer tests for $p_n<1,000,000\;(n\le 78,498)$:

Computer tests for $p_n<100,000\;(n\le 9,592)$:

What is known about this?
Is the conjecture possible to prove?

Answer 1

We know that exists a prime between $x$ and $x\left(1+\frac{1}{25\log\left(x\right)^{2}}\right)$ for all sufficiently large $x$ so if we fix $a$ we have that exists a sufficiently large $N$ such that $\forall n\geq N$ exists a prime $p$ such that $$ ap_{n}<p<\left(a+\frac{a}{25\log\left(ap_{n}\right)^{2}}\right)p_{n}$$ Now if $n$ is large enough we have (we have fix $a$ ) $$\frac{a}{25\log\left(ap_{n}\right)^{2}}<1$$ so$$ap_{n}<p<\left(a+\frac{a}{25\log\left(ap_{n}\right)^{2}}\right)p_{n}<\left(a+1\right)p_{n}.$$

Answer 2

This is an immediate consequence of the Prime Number Theorem.

Actually, the following stronger statement is true:

For each $a>1$ there exists some $N$ so that for all $n>N$ there exists a prime between $n$ and $an$.

The proof is the following: By PNT we have $$lim_n \frac{p_{n+1}}{p_n}=1$$

Therefore, there exists some $M$ so that for all $n >M$ we have $$\frac{p_{n+1}}{p_n}<a$$

Let $N=p_{M+1}$. Then, for each $n >N$ pick $p_k$ to be the last prime such that $p_{k} \leq n$. Then $p_{k+1} >n$ and $$\frac{p_{k+1}}{p_k} \leq a \Rightarrow p_{k+1} \leq a p_k \leq an$$

Answer 3

Less an answer, more toward more information. Ramanujan primes are another generalization of Bertrand's conjecture. These primes have also been generalized. There are at least 10 papers on both Ramanujan primes and its generalizations, So I would suggest that you use Google Scholar to fine them. You may find some similar numbers too.