Let $p_n$ be the $n$-th prime.

For each integer $k \ge 0$, do there exist
an infinite number of $k+3$ consecutive primes
$(p_n, p_{n+1}, \ldots, p_{n+2+k})$
so that

* (1) The gap between the 1st and 2nd, and between the
2nd and last, are equal: $p_{n+1}-p_n = p_{n+2+k}-p_{n+1}$.

* (2) There are $k$ primes between the 2nd and last,
i.e., between $p_{n+1}$ and $p_{n+2+k}$.

For $k=0$, the answer is *Yes* by the recent breakthroughs on prime gaps.
Here are some examples for $k>0$:
<br /> &nbsp; &nbsp; &nbsp;![PrimeGapsk][1]
<br />
One could whimsically imagine "skipping" a flat stone on the primes,
where the first bounce covers the gap between
the 1st and 2nd primes, followed by $k+1$ smaller
bounces that
together cover the same gap before sinking on the last prime.


  [1]: https://i.sstatic.net/5yl8h.png