Recently I was reading one of my earlier posts. There it has been conjectured that,
For all sufficiently large $x,y$ we have, $$\pi(x)+\pi(y)\le 2\pi\left(\dfrac{x+y}{2}\right)$$
But it turned out to be false as shown there. Here $\pi(x)$ denotes the number of primes less than or equal to $x$.
My question is,
Is it true that for all sufficiently large $x,y$ we have either $\pi(x)+\pi(y)< 2\pi\left(\dfrac{x+y}{2}\right)$ or $\pi(x)+\pi(y)> 2\pi\left(\dfrac{x+y}{2}\right)$? In other words, does the equation, $$\pi(x)+\pi(y)=2\pi\left(\dfrac{x+y}{2}\right)$$has only finitely many positive integral solutions?
It seems to me that the answer to both questions are "yes" but I don't know how to prove this. Can anyone help?
No, the number of integer solutions is infinite: For all $x \in \mathbb{N}$ you have $$\pi(x)+\pi(x)=2\pi\left(\frac{x+x}{2}\right)$$