Let   $p\_1\ p\_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p\_k;\ p\_n)$   by the number of primes in the double interval   $(p\_k\!+p\_{k+1};\ p\_{n-1}\!+p\_n)$; possibly by   $(2\cdot p\_k; 2\cdot p\_n)$   too. Let me ask two open questions along this line. The first one will be most likely hopeless while the second one may lead to a discussion and at least to numerical computations.


**P1.**   Does there exist a natural number   $d$   such that for every natural number   $n$   the real interval


$$ (2\cdot p\_n;\ 2\cdot p\_{n+d})$$


contains at least one prime?


**P2.**   (when **P1** fails): Given a natural number   $d$,   let   $w(d)$   be the least natural number such that the interval of **P1** (see above) does not contain any prime number. What is the growth of the sequence


$$w(1)\ \ w(2)\ \ w(3)\ \ldots$$




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The above notions got shifted from my original definition by a half of a prime. The question **Q1** below is still equivalent to question **P1** above:


**Q1.**   Does there exist a natural number   $d>1$   such that for every natural number   $n$   the real interval


$$ (p\_n\!+p\_{n+1};\ p\_{n+d-1}\!+p\_{n+d})$$


contains at least one prime?


**Q2.**   (when Q1 fails): Given a natural number   $d>1$,   let   $v(d)$   be the least natural number such that the interval of **Q1** (see above) does not contain any prime number. What is the growth of the sequence


$$v(1)\ \ v(2)\ \ v(3)\ \ldots$$


**EXAMPLE**   Consider the consecutive primes


$$p\_{360} = 1901 \qquad p\_{361}=1907 \qquad p\_{362}=1913$$


Then the real interval


$$(p\_{360}\!+p\_{361};\ p\_{361}\!+p\_{362})\ \ =\ \ (3802; 3820)$$


contains no primes, i.e.   $v(2)\le 360$.




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In general, I'd be interested in similar relative properties of primes, where primes are studied in relations to other primes, and the relation is not trivial, meaning not reduced to general properties between integers.



An interaction between prime numbers
The answer to P1 is negative, thanks to the [recent work of Maynard](http://arxiv.org/abs/1311.4600) on bounded gaps between primes.


What Maynard shows is that given any $d$, there exists a k-tuple $h\_1,\dots,h\_k$ such that for infinitely many $n$, at least $d+1$ of $n+h\_1,\dots,n+h\_k$ are prime. In fact, the argument shows that for sufficiently large $x$, the number of $n \in [x,2x]$ such that at least $m$ of $n+h\_1,\dots,n+h\_k$ are prime, and the rest are almost prime in the sense that they have no prime factor less than $x^\varepsilon$ for some small fixed $\varepsilon>0$, is $\gg \frac{x}{\log^k x}$. (Such strengthenings of Zhang/Maynard type theorems are discussed in [this paper of Pintz](http://arxiv.org/abs/1305.6289), and also in [this Polymath8b preprint](https://www.dropbox.com/sh/0xb4xrsx4qmua7u/tfwv3_O_WY/Polymath8b/newergap.pdf).) A standard upper bound sieve (e.g. Selberg sieve or beta sieve) then shows that after removing about $O( x/\log^{k+1} x)$ of these $n$, one can also ensure that none of the numbers between $2(n+h\_1)$ and $2(n+h\_k)$ are prime. If we let $p\_i$ be the first prime greater than or equal to $n+h\_1$, then we have $2(n+h\_1) \leq 2p\_i \leq 2p\_{i+d} \leq 2(n+h\_k)$, and so we obtain a counterexample to P1 for any $d$.


Unfortunately we don't get an effective rate on P2 this way due to the reliance on the Bombieri-Vinogradov theorem in the work of Maynard etc. However this can likely be removed by following the ideas mentioned near the end of [this paper of Pintz](http://arxiv.org/abs/1305.6289), though I did not attempt this. [EDIT: it looks likely that the quantitative version of Maynard's results in [this recent paper of Banks, Freiberg, and Maynard](http://arxiv.org/abs/1404.5094) will do the trick, after some small modification.]


