Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:15:08Z http://mathoverflow.net/feeds/question/71639 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71639/conjectureif-ij-then-pipii-i-pipjj-j1 Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$ a-boy 2011-07-30T06:15:12Z 2011-07-30T08:09:27Z <p>p[i] is the i-th prime. $\pi(x)$ is prime counting function.</p> <p>Firstly, I think that this Prime gap inequality holds true, </p> <p>$p[i+1] - p[i] &lt;= i$</p> <p>Prove:for any i>0, we can always find distinct prime factors for {p[i], p[i]+1,...,p[i+1]}. For example, i=11, p[11]=31, p[12]=37, {31,32,33,34,35,36,37} have distinct prime factors {31,2,11,17,5,3,37}. Pigeonhole principle shows this simple inequality! </p> <p>My question is the title,</p> <p>Conjecture(Prime counting inequality) :</p> <p>if $i\lt j$, then $\pi(p[i]+i)-i&lt;=\pi(p[j]+j)-j+1$</p> <p>Edit: sorry, this conjecture is false! But another question is arising: if $i\lt j$, what is the max value of $g(i,j) = (\pi(p[i]+i)-i) - (\pi(p[j]+j)-j) /; i\lt j$</p> <p>I find g(i,j) may be 12. g(150065,150090)=12.</p> <p>and more, what is the max value of h(i,j)=j-i /; $(\pi(p[i]+i)-i) > (\pi(p[j]+j)-j)$</p> http://mathoverflow.net/questions/71639/conjectureif-ij-then-pipii-i-pipjj-j1/71642#71642 Answer by Gerhard Paseman for Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$ Gerhard Paseman 2011-07-30T07:03:45Z 2011-07-30T07:03:45Z <p>I believe your inequality $p(i+1) \leq p(i) + i$ is true, but that there is no short and elementary proof. It follows from inspection and some known results on the length of gaps between primes, cf. Dusart or Harman.</p> <p>Your titled inequality I believe fails for some $i$ with $j=i+2$. I don't have a specific value for $i$, but I suspect that there is a sequence such that your arguments to $\pi$ could fall into a large gap, especially when $p(i+2)= p(i)+6$. That is where I would start looking for counterexamples.</p> <p>Gerhard "But It Looks Quite Plausible" Paseman, 2011.07.30 </p>