# Simultaneous solutions of two inequalities involving prime numbers

Are there infinitely many solutions to the following inequalities (with $$x\ne y$$ and $$x+y$$ odd),

$$x+y>p_{\pi(x)}+p_{\pi(y)+1}\tag{1}$$and $$x+y>p_{\pi(x)+1}+p_{\pi(y)}\tag{2}$$where $$\pi(x)$$ denotes the number of primes less than or equal to $$x$$?

In one of my previous post it has been shown that the inequality, $$x+y<\dfrac{p_{\pi(x)}+p_{\pi(y)+1}+p_{\pi(x)+1}+p_{\pi(y)}}{2}$$fails infinitely often. So, one can expect that there are simultaneous solutions to $$(1)$$ and $$(2)$$ but I can't prove it. Can anyone help me?

Yes. Let $p_k\ge p_{k-1}+4$ be any prime following a gap of at least 4 and let $x=p_k-1,y=p_k-2$. Then $$x+y = 2p_k-3 \\ p_{\pi(x)}+p_{\pi(y)+1} = p_{\pi(x)+1}+p_{\pi(y)} = p_k+p_{k-1} \le 2p_k-4$$