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Let $\pi(x)$ denotes the number of primes less than or equal to $x$. Also suppose that for some fixed $N$ we have $\pi(x+y)\ge\pi(x)+\pi(y)$. The problem is,

Show that the equality in the above inequality cannot hold for all $x,y>3$ and $x+y\le N$ where $x$ and $y$ are both positive integers.

My attempt

Suppose that there exists some $x_0$ and $y_0$ such that we have $$\pi(x_0)+\pi(y_0)=\pi(x_0+y_0)$$ Now if $x_0+y_0$ is composite then $\pi(x_0+y_0)=\pi(x_0+y_0-1)$. Now note that none of $x_0$ and $y_0$ can be prime because then we will have a contradiction. So the essence of all this is that if $x_0+y_0$ is composite then we also have a solution of the equation $$\pi(x+y-1)=\pi(x’)+\pi(y’)$$ such that $x’+y’=x+y-1$.

But I cannot solve the case when $x_0+y_0$ is a prime. Can anyone help me?


Added

After going through the answer below, I think that I should elaborate once more the problem. What it says in disguise (so far I have understood) is the following,

Prove that if $p$ be an odd prime then there doesn't exist any solution to the equation, $$\pi(p-1)=\pi(x)+\pi(p-1-x)$$ for any integer $x$ where $p-x$ is a composite number.

Hope that this clarifies the problem more. If there is any problem (e.g., vagueness) still remaining, please let me know.

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    $\begingroup$ What do you mean when you say "for some fixed $N$ we have" a claim that doesn't mention the letter $N$ at all? $\endgroup$ – Henning Makholm May 23 '15 at 14:23
  • $\begingroup$ Probably it means that for some particular $N$ and $x$ there is a solution to the equality. $\endgroup$ – user 170039 May 25 '15 at 5:01
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The following solution works for all $N \geq 9$.

Let $y=5$ and $x=N-5$. Then we should have $$\pi(N) \geq \pi(N-5) + \pi(5) = \pi(N-5) + 3$$

Hence, at least three of $N-4$, $N-3$, $N-2$, $N-1$, and $N$ must prime. That is not possible unless $N-4=2$ or $N-4=3$, in which case we have $N \leq 7$. Contradiction.

Note: We have $N \geq 9$ so that $x=N-5 \geq 4$.


Further, since $x,y > 3$, we have $x+y=N \geq 8$. So we only have $x=y=4$ in the case $N=8$, and your statement isn't even true, because we have $$\pi(8) = 4= 2\cdot 2 = 2\cdot \pi(4)$$

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  • $\begingroup$ So, it implies that if we assume $\pi(x_0)+\pi(y_0)=\pi(x_0+y_0)$ holds for some $x_0,y_0$ where $x_0+y_0$ is a prime, then we can apply induction to reduce the statement to obtain a contradiction in the manner you said, right? To elaborate suppose there exists solution of the equation $\pi(x')+\pi(y')=\pi(x+y)$ with $x'+y'=x+y$ then we assume that there exists solution of the equation $\pi(x')+\pi(y')=\pi(x+y-r)$ where $x'+y'=x+y-r$. Then to prove the existence of the solution of $\pi(x')+\pi(y')=\pi(x+y-r-1)$ we apply your argument and eventually reach a contradiction. Is the process right? $\endgroup$ – Falling Snow May 24 '15 at 5:58
  • $\begingroup$ Also just one more question, the assumption is that for some $x_0$ and $y_0$ we have $\pi(x_0+y_0)=\pi(x_0)+\pi(y_0)$ with $x_0$ and $y_0$ fixed. How do you use this statement explicitly? $\endgroup$ – Falling Snow May 24 '15 at 6:05
  • $\begingroup$ No, I didn't use your attempt. I just showed, by providing counterexample, that it isn't even possible that for all $x,y$ to have this inequality. Maybe I misunderstood the problem? It is vaguely formulated. $\endgroup$ – wythagoras May 24 '15 at 6:29
  • $\begingroup$ Please see the added part. $\endgroup$ – Falling Snow May 24 '15 at 15:24

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