4 added 271 characters in body

p[i] is the i-th prime. $\pi(x)$ is prime counting function.

Firstly, I think that this Prime gap inequality holds true,

$p[i+1] - p[i] <= i$

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!

My question is the title,

Conjecture(Prime counting inequality) :

if $i\lt j$, then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

Edit: sorry, this conjecture is false! But another question is arising: if $i\lt j$, what is the max value of $g(i,j) = (\pi(p[i]+i)-i) - (\pi(p[j]+j)-j) /; i\lt j$

I find g(i,j) may be 12. g(150065,150090)=12.

and more, what is the max value of h(i,j)=j-i /; $(\pi(p[i]+i)-i) > (\pi(p[j]+j)-j)$

3 deleted 21 characters in body

p[i] is the i-th prime. $\pi(x)$ is prime counting function.

Firstly, I think that this Prime gap inequality holds true,

$p[i+1] - p[i] <= i$

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!

My question is the title,

Conjecture(Prime counting inequality) :

if $i\lt j$, then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

Edit: sorry, this conjecture is flasefalse! I will del this post.

2 added 66 characters in body

p[i] is the i-th prime. $\pi(x)$ is prime counting function.

Firstly, I think that this Prime gap inequality holds true,

$p[i+1] - p[i] <= i$

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!

My question is the title,

Conjecture(Prime counting inequality) :

if $i\lt j$, then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

Edit: sorry, this conjecture is flase! I will del this post.

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