Legendre's Conjecture Theme (Part I)

Question

Main Question

Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following,

Conjecture 1. For all sufficiently large $n\in\mathbb{N}$ we have, $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)\ne2\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)$$

Conjecture 2. If, $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)=2\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)$$then either, $$\pi\left((n+1)^2\right)-\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)>0$$ or, $$\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)-\pi\left(n^2\right)>0$$ for all sufficiently large $n\in\mathbb{N}$.

To me the second conjecture seems more difficult. However, my questions are,

1. Is there a proof of any one of the conjectures stated above? If not, (which is highly probable), can some literature be mentioned which investigates positive integral solutions of any one of the following equations, $$\pi\left(x\right)+\pi\left(y\right)=a\pi\left(\dfrac{x+y}{a}\right)$$or, $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)=a\pi\left(\dfrac{(n+1)^2+n^2}{a}\right)$$ where $a>1$.

2. If the answer of my first question is negative then can anyone give me some hint as to how I should go about proving my any one of the conjectures?

3. Has there been any attempt towards resolving the conjecture by examining the solutions of $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)=2\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)$$

Remarks (for the sake of completeness)

• As to the question of how one can prove Legendre's Conjecture using any one of the conjectures, we discuss only the proof of Legendre's Conjecture assuming Conjecture 1 (although admittedly it is trivial). The proof of Legendre's Conjecture assuming Conjecture 2 is also trivial and not discussed.

  Case 1. Let $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)>2\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)$$In this case observe that since $\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)-\pi\left(n^2\right)\ge 0$, we get that, $$\pi\left((n+1)^2\right)-\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)>0$$and we are done in this case.

  Case 2. Let $$\pi\left((n+1)^2\right)+\pi\left(n^2\right)<2\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)$$In this case observe that since $\pi\left((n+1)^2\right)-\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)\ge 0$, we get that, $$\pi\left(\dfrac{(n+1)^2+n^2}{2}\right)-\pi\left(n^2\right)> 0$$and we are done in this case.

• In fact using the same argument we can prove the following theorem,

  Theorem. Let $f:D(\subseteq\mathbb{R})\to\mathbb{R}$ be a function such that $f(n)$ is strictly increasing. Then there exists a prime between $f(n)$ and $f(n+1)$ if any one of the following holds,

  1. $\pi\left(f(n+1)\right)+\pi\left(f(n)\right)\ne2\pi\left(\dfrac{f(n+1)+f(n)}{2}\right)$ for all sufficiently large $n\in\mathbb{N}$.

  2. If $\pi\left(f(n+1)\right)+\pi\left(f(n)\right)=2\pi\left(\dfrac{f(n+1)+f(n)}{2}\right)$ then for all sufficiently large $n\in \mathbb{N}$ we will have either $\pi\left(f(n+1)\right)-\pi\left(\dfrac{f(n+1)+f(n)}{2}\right)>0$ or $\pi\left(\dfrac{f(n+1)+f(n)}{2}\right)-\pi\left(f(n+1)\right)>0$.

Further Added

A continuation of this question has been asked here where an alternative conjecture is proposed which would aid us in proving the Legendre's Conjecture even if the conjectures mentioned here doesn't hold in general.