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I've asked the same question on MathOverflow two days ago as On a conjecture about the arithmetic function that counts the number of twin primes, I add this reference while I hope to know what about my MathOverflow post.

In this post we denote (for a fixed positive integer or real number $x$) as $$\pi_2(x)=\#\{\text{ primes }p\leq x\,:\,p+2\text{ is also a prime}\}$$ the arithmetic function that counts the number of primes $p$ less than a given positive real $x$ satisfying that $p+2$ is also a prime. As general reference I add the article from Wikipedia Twin prime that refers that is unproven the existence of infinitely many twin primes, and the articles also from Wikipedia Second Hardy–Littlewood conjecture. I was inspired in these articles and a few experiments using Pari/GP scripts to state the following conjecture.

Conjecture 1. One has $$\pi_2(x+y)\leq \pi_2(x)+\pi_2(y)+1$$ for all integer $x\geq 2$ and all integer $y\geq 2$.

Conjecture 2. There exists an integer constant $K\geq 2$ such that $$\pi_2(x+y)\leq \pi_2(x)+\pi_2(y)+1$$ holds, for all integer $x\geq K$ and all integer $y\geq K$.

Question. Are these known, or is it possible to prove or refute any of previous conjectures? Can you find counterexamples for the first conjecture or add heuristics to know what about the veracity of this kind of conjectures? Many thanks.

I've tested the first conjecture for the segments of integers $2\leq x,y\leq 500$. I've added the second conjecture since I don't know if it is possible to find easily a counterexample for Conjecture 1.

Edit: Additionally you can investigate what about the veracity of the conjectures on assumption that the Twin Prime Conjecture holds. See the comments, many thanks to this user for this contribution.

References:

[1] G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. (44), J. E. (1923) pp. 1–70.

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    $\begingroup$ See for a related question. $\endgroup$ – Dietrich Burde Mar 24 at 14:28
  • $\begingroup$ A perfect feedback and many thanks for your attention @DietrichBurde $\endgroup$ – user759001 Mar 24 at 14:29
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    $\begingroup$ $500$ isn't a very large number...if I recall correctly, the equivalent conjectures for the ordinary $\pi(x)$ are inconsistent with the conjectured asymptotics for prime $k$-constellations (I may have that wrong). Are these conjectures consistent with those? $\endgroup$ – lulu Mar 24 at 14:35
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    $\begingroup$ Note that if there are only finitely many twin primes then Conjecture 2 definitely holds. So, the problem is to investigate only the case when the Twin Prime Conjecture holds. $\endgroup$ – user170039 Mar 29 at 4:49
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    $\begingroup$ Play with prime constellations? For example, knowing we can't have $3$ consecutive primes other than $(3,5,7)$ among $x$ consecutive numbers, implies we can have at most $1$ new lesser twin prime ($p$ s.t. $p+2$ is also prime) every $6$ numbers, for $x\gt 3$. This gives: $\pi_2(y+x)\le \pi_2(y)+\lfloor x/6 \rfloor$. Checking $\lfloor x/6 \rfloor \le \pi_2(x)+1$ holds for $3 \lt x\le 47$ we already have the conjecture holds for all $(x\le 47, y\in\mathbb N)$ since $x\le 3$ are easy to verify. Try improving $x$ with longer constellations? Or do asymptotics imply a counterexample for large $x$? $\endgroup$ – Vepir Mar 29 at 10:30

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