# What about the question "Hint for the Elementary Proof of the Inequality of Prime Numbers"?

I think that my question (actually it is not originally my question, it is from someone on the AoPS) might suggest us to prove the Bertrand's Postulate in an elementary way (only if my question can be proved in an elementary way). But unfortunately I found that it has been downvoted and also labelled as "off-topic". Can anyone tell me the precise reason why it is considered as "off-topic"?

• I don't know what the downvote signifies; my guess is that it means "this question is not welcome here." However, it's an honest question, and I was the one who suggested that the OP come here for enlightenment.
– Todd Trimble Mod
May 12, 2014 at 5:37
• An observation: you significantly changed the question (after closure) without any explanation. This is a bit strange.
– user9072
May 12, 2014 at 19:01
• I have changed the question just because I found a better inequality from which the earlier inequality would follow. Note that my earlier post was also taken from AoPS and it was also posted by the same author.
– user48849
May 13, 2014 at 7:50
• I see that on MSE you got an answer for your problem along the lines I sketched below. Since you seem unsatisfied with that too (from your comment to that answer, and mine), my suggestion would be to ask the AoPS proposer what he/she had in mind. Quid made the interesting point that your original question (which was really the one that was closed) was different and much easier. It could be that the AoPS proposer intended people to be able to solve that question, and the harder problem was to get people to think more and be challenged. I definitely think you should ask the AoPS proposer. May 13, 2014 at 14:07
• The question has now been (perfunctorily) edited, and as I said below I have voted to reopen it. Others can judge whether it really merits reopening or not. May 14, 2014 at 15:12

I was one of those who voted to close -- the problem seemed unmotivated and simply presented as something seen on AoPS. Here are some suggestions on perhaps reformulating the question, and on the math itself. If you'd like to edit your question, I'm happy to give the benefit of doubt and will vote to reopen. But perhaps what I say here will be enough for you.

The question is better formulated as "Is there a prime in the interval $[n,n+\pi(n)]$ for every $n\ge 2$?" Note that this is a good deal stronger than Bertrand's postulate asking for a prime in $[n,2n]$, which anyway has a simple elementary proof. Your question follows easily for large $n$ by the prime number theorem with a strong error term; possibly we know enough to check it for all $n$. Since the prime number theorem with a strong error term has an elementary proof (in the sense of avoiding complex analysis), so does your question. But perhaps what you want is a simple proof; this may not be easy.

Anyway the problem as stated seemed unsuitable to me, and even with the reformulation I think it is borderline, but as I said I'd be happy to give the benefit of doubt.

• That's a much clearer formulation (for me) -- thanks, Lucia. I'm not terribly familiar with AoPS, and so I don't know whether problems are posted with the idea that they are known to have simple, elegant solutions, or if all manner of questions are posted, some perhaps not meeting this criterion.
– Todd Trimble Mod
May 12, 2014 at 18:46
• @ToddTrimble I am not very familiar with that site either but: I am under the impression they have various categories and this one was posted in 'Number Theory Proposed & Own problems' without any source (often sources are given). Thus it seems likely this was created by the person posting there. If one would want to know something about the problem one might ask them. It was posted there only two days ago. Generally speaking I completely fail to see the point of somebody reposting a problem from there here.
– user9072
May 12, 2014 at 19:20
• Even using results of Dusart, I would not call the derivation elementary. May 12, 2014 at 21:56
• @TheMaskedAvenger: I don't understand your comment. I clarified that the prime number theorem with a strong error term (main terms $+O(x/(\log x)^3)$) say, could be proved elementarily in the sense of avoiding complex analysis, but not in the sense of being simple. May 12, 2014 at 22:04
• @Lucia, I will explain it in detail if you wish (but I don't think it will help you much). However, I made it for the benefit of Todd Trimble, to suggest that even if the original poster were given Dusart's recent results, the derivation of "there is a prime in between x and x +pi(x)" from that I wohld not consider as elementary. May 13, 2014 at 0:41
• @Lucia: Note that the problem was posted in the Olympiad Section. The author may have wanted that the problem to be solved by undergraduate tools and so he (or she) posted it to AoPS. In fact I mentioned the name of AoPS only to stress this aspect. If you have seen the link that I have given regarding my progress on the proof, then you must have noticed that I have only mentioned some trivial elementary results as my 'progress'. If I would intend to use analytic approach I could do so but if the problem would have a 'simple' solution it would be quite fascinating.
– user48849
May 13, 2014 at 7:49
• @TheMaskedAvenger My benefit? I don't quite follow. I never used the word 'elementary' (which has a technical meaning already mentioned by Lucia). I'm sure you're right that no simple solution exists; the main beneficiary of your insight would probably be William Hilbert.
– Todd Trimble Mod
May 13, 2014 at 20:54
• Who is William Hilbert? Aug 5, 2020 at 22:32