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I asked this question yesterday, perhaps a bit too hastily:

Does the prime number theorem tell us that the next prime number is guaranteed to be relatively nearby?

I think I bit off more than I can chew with this question, as I know little about the maths surrounding the prime number theorem. However, I asked the question assuming that any proof that

$$ \displaystyle\lim_{n\to\infty}\ \frac{p_n}{p_{n+1}} = 1\quad \text{where}\ p_n\ \text{is the n-th prime number}?$$

will require knowledge of the PNT, especially the proof of the PNT itself, which I do not have. Is there an elementary proof of the above statement, that does not require knowledge of the PNT or related facts, or difficult number theory?

For example, the proof that there are infinitely many primes is not too difficult to understand, and doesn't require any advanced number theory. So I am wondering if there is a proof that the primes are eventually relatively close with a similar level of difficulty to the proof that there are infinitely many primes. Sorry for not providing this information sooner.

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    $\begingroup$ Off hand. I doubt it. $\endgroup$ May 22, 2021 at 17:01
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    $\begingroup$ Why, because that's the whole point of the PNT and the surrounding maths? $\endgroup$ May 22, 2021 at 17:02
  • $\begingroup$ It is a corollary of PMT, although not the only point. $\endgroup$ May 22, 2021 at 17:07
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    $\begingroup$ I think Bertrand's "postulate", which says there's always a prime between $n$ and $2n$, is a non-trivial theorem. $\endgroup$ May 22, 2021 at 17:35
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    $\begingroup$ @Peter " "related facts" - a quite broad formulation what is allowed and what not. " That's a good point. I guess I mean- not containing any difficult number theory. It sounds like the answer is "no", especially due to Andreas Blass's comment: if PNT were easy to prove/"trivial" then Bertrand's Postulate would also be "trivial". So there we go. $\endgroup$ May 22, 2021 at 17:43

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The first elementary proof of the PNT famously happened when Erdos proved that $\lim_{n\to\infty}\frac{p_n}{p_{n+1}}=1$, and then Selberg combined that with some of his other works to derive the PNT in an elementary fashion. If you wish to call Erdos's proof of the $\lim_{n\to\infty}\frac{p_{n}}{p_{n+1}}=1$ an "Elementary proof without prime number theorem related maths" then you have your answer, but seeing as this was the crucial step missing for a full elementary PNT proof I'd call that a stretch. I don't think that there is any more complete answer to your question one can give, short of transcribing a rendition of Erdos's proof.

An account of the first elementary proof of the PNT is here. Be warned, however. What Erdos and Selberg call elementary might not look so elementary to you.

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An elementary, if not 100% rigorous proof would be a prime number density function. There are several prime number density functions of increasing complexity, but using the simplest, the density of primes is about log N. (Natural Log of N). Where N is the starting number. So, most of the time, there is a prime within log N of any number.

If you could RIGOROUSLY prove how close the next prime is to ANY number, and get a number that is around C * log N, with a modest value for C. Then, I would expect such a proof could be used to complete the GOLDBACH conjecture.

Additionally, the entire number line from 2-2**64 has been searched. There are no non-trivial examples where the gap is > (log N)**2.

Max Gap 1132 1693182318746371 1693182318747503 has a gap of 0.92 * (log N)**2

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