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Source. Lemma 2.1 (p. 4) and Theorem 2.2 (p. 5), Section 2, of Nathan McNew, The convex hull of the prime number graph, in: Irregularities in the Distribution of Prime Numbers, Springer, Cham (2018), 125--141, doi:10.1007/978-3-319-92777-0_7, cited at the page numbers 1--15 of the author's preprint named on the source card.

Statement

Setting (pp. 1--3). The prime number graph is the set of points $(n,p_n)$, $p_n$ the $n$th prime. A convex prime is a prime $p_n$ for which $(n,p_n)$ is a vertex of the convex hull of this graph; $c_1<c_2<\cdots$ are the indices of the convex primes, so the convex primes are $p_{c_1}<p_{c_2}<\cdots$.

Lemma 2.1 (p. 4). If $(m,p_m)$ is any point on the boundary of the convex hull of the prime number graph, the segment of the hull boundary following it has slope $\log m+\log\log m+o(1)$ as $m\to\infty$.

Theorem 2.2 (p. 5). The number of convex primes up to $x$ is

$$O\Bigl(\frac{x^{2/3}}{\log^{2/3}x}\Bigr).$$

The paper notes (p. 5) that this proves Tutaj's Conjecture 1.2 (p. 3), that $\sum_{i\ge1}1/p_{c_i}$ converges. It also notes (p. 3) that the bound is $O(\pi(x)^{2/3})$, which improves the earlier $o(x/\log x)$ that Pomerance drew from a result of Erdős and Prachar.

Read depth. Claims checked: Lemma 2.1 and Theorem 2.2 were read clause by clause on the page images of the preprint; the proofs were read but not checked, and nothing here is independently reviewed.

Proof pointer

p. 5. Count the convex primes in $(\tfrac12x,x]$. The slopes between consecutive convex primes are strictly increasing rationals $(p_{c_{j+1}}-p_{c_j})/(c_{j+1}-c_j)$, and by Lemma 2.1 they lie in an interval of length $\log2+o(1)$, so for each index gap $k$ there are $O(k)$ possible slopes. Consecutive convex primes with index gap at most $K$ therefore number $O(K^2)$, those with gap above $K$ number $O(x/(K\log x))$, and $K=(x/\log x)^{1/3}$ balances the two; then sum dyadically. The paper also records (p. 5) that the bound follows from Andrews's $O(A^{1/3})$ bound on the vertices of a convex lattice region of area $A$.

Dependencies

Lemma 2.1 (above) and the prime number theorem.

Bears on

No Erdős problem directly. The convex primes are a subset of the midpoint convex primes, the primes with $M_n>0$ in the notation of equation (23); an upper bound on how many convex primes there are says nothing about how large $M_n$ can be, which is what Problem 454 asks.