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Bertrands postulate states that there's always a prime number in [N,2N] and I was thinking...

Considering that N=1*N and that (1,2) are the first prime numbers maybe this is just a particular case and there's a more general law such as:

"There is always a prime between hN and pN for every couple of prime numbers (h,p) with h<p"

I made some scripting and tested it for the first 1000 numbers, it turns out it could be the case, but it applies only to h,p <= to N.

For h,p larger than N there seem to be gaps when h-p is small compared to the values of the intervals [hN,pN].

Has anybody already looked into this? Did I tap into something new? test

Fabio Beka
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    1 is not a prime! – Brauer Suzuki May 22 '22 at 17:24
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    There are arbitrary large ranges of integers not containing a prime. Namely, $n!+2,n!+3,\ldots,n!+n$ – Brauer Suzuki May 22 '22 at 17:29
  • Ok, so how about the test I conducted? Is there any reason to consider it invalid? if you open the image at the bottom you'll se the code and results. – Fabio Beka May 22 '22 at 17:41
  • The Prime Number Theorem implies that if $0<\alpha<\beta$ then for all $N$ sufficiently large there is a prime between $\alpha N$ and $\beta N$. – Gerry Myerson May 23 '22 at 05:31
  • @GerryMyerson And does this theorem say anything about and being prime numbers? Do you think the experiment I made could have any value in relation to the theorem you mentioned? – Fabio Beka May 23 '22 at 08:33
  • No, Fabio, the theorem doesn't even require $\alpha$ and $\beta$ to be integers, just arbitrary reals (so long as $0<\alpha<\beta$). One can ask how big $N$ has to be as a function of $\alpha$ and $\beta$, and your experiments may provide some data points, but I think people have looked at this sort of question a lot over the 100-plus years of the Prime Number Theorem and probably already have some fair ideas about what the answers may be. – Gerry Myerson May 23 '22 at 13:39
  • @GerryMyerson Are you able to guide me towards some texts or forums where I can check whether this has already been discovered? I would like to see whether there's also a theoretical justification. – Fabio Beka May 23 '22 at 20:57
  • The Prime Number Theorem is in every Analytic Number Theory textbook ever written, and in many intro Number Theory texts as well, and the result I state follows from it. Discussion of the result, I'll have to get back to you when I've had a chance to consult my library. – Gerry Myerson May 24 '22 at 02:13
  • Hardy and Wright, An Introduction to the Theory of Numbers, 6th edition, page 494, shows, as a simple consequence of the Prime Number Theorem, that for any $\epsilon>0$ there exists $x_0=x_0(\epsilon)$ such that for all $x>x_0$ there is a prime $p$ satisfying $x<p<(1+\epsilon)x$. – Gerry Myerson May 25 '22 at 00:33

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If $h$ and $p$ are large twin primes and $N=2$, this is rarely true. Take for instance $(h,p)=(59,61)$.

Brauer Suzuki
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  • I know, I noticed that, I wrote it in my question. But with N larger than p,h it seems like the rule seems to hold, is this something that has already been looked into or did I just tap into something new? – Fabio Beka May 23 '22 at 08:29
  • Sorry, I overlooked this (because it is hard to read without proper tex) – Brauer Suzuki May 23 '22 at 13:05