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I have encountered some error in the details of what Legendre's conjecture implies about bounded prime gaps. So I am working to correct errors and to state both what is conjectured and what is implied by the conjecture concerning bounded prime gaps. I am asking about the correctness of my corrections.

First, as to what is conjectured, because 3 is the only prime of the form $n^2-1$, I state the conjecture thus: In the interval between $n^2$ and $n^2$+2$n$, there will always be at least one prime number.

Second, as to what this conjecture implies about bounded prime gaps, let $n$ be any given natural number, let $p$ be the next prime greater than $n$, and let $m$ be the limit from $n$ to $p$ such that $n$+$m$ $\geq p$. Legendre's conjecture implies that the prime gap above $n$ can be bounded by the product of two factors, one of them being a constant and the other one being related to the square root of $n$. Let $c$ be the factor which is a constant and let $f$ be the factor which is related to the square root of $n$ such that $cf$=$m$.

My question here is about the correctness of what I have stated here so far before I attempt to find the precise relationships between the things mentioned herein which I have represented by $n,p,m,c,$ and $f$.

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  • $\begingroup$ The error which I encountered is actually one that I made. I think I understand this matter now. $\endgroup$ – Jeffrey Young Oct 3 '14 at 22:45
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Yes, Legendre's conjecture that there is a prime in $[n^2, (n+1)^2]$ ($n$ being a positive integer) is equivalent to the conjecture that there is a prime in $[n^2+1, (n+1)^2-2]$.

Legendre's conjecture doesn't imply bounded gaps, but it does imply that the gap following a prime $p$ is $O(\sqrt p)$. In particular it is at most $4\sqrt{p-1}$. Of course everyone knows in their hearts that much more is true, but that's all we get from Legendre's conjecture.

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  • $\begingroup$ So I made the error which I was seeking to correct. $\endgroup$ – Jeffrey Young Oct 3 '14 at 16:37
  • $\begingroup$ So if $n$ + 4$\sqrt n-1$ $\geq p$, where $n$ is any given natural number and $p$ is the next prime after $n$, then Legendre's conjecture is true, right? $\endgroup$ – Jeffrey Young Oct 3 '14 at 16:51
  • $\begingroup$ @JeffreyYoung: No. (Try it with $n=9$ to see why it fails.) But if you can get the gap down to $n+\sqrt n-1$ then it should work. The approximate factor of 4 between these two is a measure of the cost of the converting between the discrete world and the continuous one. $\endgroup$ – Charles Oct 3 '14 at 19:34
  • $\begingroup$ math.stackexchange.com/questions/957101/… I asked about this in a new question. Have I got it right now? $\endgroup$ – Jeffrey Young Oct 3 '14 at 22:14
  • $\begingroup$ This matter, to me, is resolved. See the question at the link above. The matter which I was wondering about is not significant, but the answers which I received are very enlightening. $\endgroup$ – Jeffrey Young Oct 3 '14 at 23:27

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