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Source. Theorem 2.3 (p. 5) and Corollaries 2.5, 2.7 and 2.8 (p. 8), 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 $p_{c_1}<p_{c_2}<\cdots$.

Theorem 2.3 (p. 5). There is a constant $B>0$ such that for all sufficiently large $i$,

$$p_{c_{i+1}}-p_{c_i}\le p_{c_i}\exp\Bigl\{\frac{-B\log^{3/5}p_{c_i}}{(\log\log p_{c_i})^{1/5}}\Bigr\}.$$

Corollary 2.5 (p. 8). There is a constant $B>0$ such that the number of convex primes up to $x$ is at least

$$\exp\Bigl\{\frac{B\log^{3/5}x}{(\log\log x)^{1/5}}\Bigr\}.$$

Corollary 2.7 (p. 8). The sum $\sum_{i\ge1}1/\log p_{c_i}$ diverges. The paper derives it from the lower bound of Corollary 2.5 and presents it as the proof of Tutaj's Conjecture 1.3 (p. 3).

Corollary 2.8 (p. 8). $\lim_{i\to\infty}p_{c_{i+1}}/p_{c_i}=1$. This is Tutaj's Theorem 1.1 (p. 3), proved by Tutaj under the Riemann Hypothesis; here it holds unconditionally, since Theorem 2.3 gives $p_{c_{i+1}}-p_{c_i}=o(p_{c_i})$.

The paper does not say that the $B$ of Corollary 2.5 is the $B$ of Theorem 2.3. Page 7 says that Corollary 2.5 settles a claim Pomerance made without proof, of a count at least $e^{c\log^{3/5-\epsilon}x}$ for some $c>0$.

Read depth. Claims checked: Theorem 2.3 and Corollaries 2.5, 2.7 and 2.8 were read clause by clause on the page images of the preprint; the proof of Theorem 2.3 (pp. 5--7) was read but not checked, and nothing here is independently reviewed.

Proof pointer

pp. 5--7. The prime number theorem with the best known error term puts every point of the graph between the two convex curves $x=\mathrm{li}(y)\mp Dy\exp\{-A\log^{3/5}y/(\log\log y)^{1/5}\}$. A boundary segment of the hull lies between them. Comparing its midpoint with the inner curve through a Taylor expansion of $\mathrm{li}$ bounds its vertical extent by a constant times $y\log y\exp\{-A\log^{3/5}y/(2(\log\log y)^{1/5})\}$, which gives the gap bound. The corollaries follow from the gap bound (pp. 7--8).

Dependencies

The prime number theorem with error term $O(x\exp\{-A\log^{3/5}x/(\log\log x)^{1/5}\})$, equation (5), p. 5.

Bears on

No Erdős problem directly. Since every convex prime is a midpoint convex prime (p. 2), Corollary 2.5 bounds from below the number of $n$ with $M_n>0$ in the notation of equation (23); it says nothing about how large $M_n$ can be, which is what Problem 454 asks.