Does the sequence of prime numbers contain contiguous subsequences of length $k$ which are strictly convex (similarly, strictly concave), for every $k\in\mathbb{N}?$ For example,
$$ 17, 19, 23, 29 $$
is contiguous and strictly convex with $k=4.$
and
$$61, 67, 71, 73$$
is contiguous and strictly concave with $k=4.$
I'm sure we can find sequences of contiguous primes with $k=5,6$ i.e. of length $5$ or $6$ for each strictly concave and strictly convex. But can we find $k=$ arbitrarily large for both strictly concave (and strictly convex)?
If we add $1$ to every member of the Thue-Morse sequence and call this sequence $(t_n),$ and define $x_1=1, x_{k+1} = x_k + t_k.$ Then then looking at the sequence $(x_n)$ shows that you can even have strictly increasing sequences of positive integers which have positive density and have a cap (upper bound) on $k,$ the longest contiguous strictly concave/convex subsequence of $(x_n),$ (which is $k=3$ due to the fact the T-M is cube-free). So basically, the density of the primes $p_n$ as $n\to\infty$ doesn't help us answer the above question. It is true, however, that the PNT tells us that the size of the largest prime gap $\to\infty$ as $n\to\infty.$ I'm not sure this helps either though.