I am sorry! $n+1 \equiv 1 $ mean: Taking subscripts modul $n$, $\ell$ is real number. I corrected.

@GerryMyerson Thank you, I have just hecked and updated

I thinks this conjecture is stronger than some old conjecture. Because $\ln x > 1, 2, 3, 4,.....$ when $x> e^{1}, e^{2}, e^{3}, e^{4}.....$ and with any positive interger $n$ then exist $x$ such that $(x-ln^2{x}, x+ln^2{x}) \subset (n^2, (n+1)^2)$

Yes, you are right. I don't know what Conjecture 1 has to do with Goldbach's conjecture, may you help me? @GerryMyerson

@GerryMyerson Today, I see that maybe conjecture 1 equivalent to Goldbach's conjecture en.wikipedia.org/wiki/Goldbach%27s_conjecture or Corollary of the Goldbach's conjecture, because $P_n+P_m=P_{n+m}-x=y$. Where $x$ is an odd positive integer less than $P_{n+m}$ then $y$ is an odd positive integer less $P_{n+m}$. But maybe we must prove that $P_n+P_m \le P_{m+n}$. Is my remark right?