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Source. Equation (1) (p. 2) and Section 5 (pp. 12--14: the table of $M(x)$, p. 12; equation (23) and Figure 2, p. 13; Questions 5.1 and 5.2, p. 14) 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

Midpoint convex primes (p. 2). Following Pomerance, a midpoint convex prime is a prime $p_n$ with

$$2p_n<p_{n-i}+p_{n+i}\quad\text{for all positive }i<n. \qquad (1)$$

Every convex prime (a prime whose point $(n,p_n)$ is a vertex of the convex hull of the prime number graph) is a midpoint convex prime (p. 2).

Equation (23) (p. 13). Put

$$M_n=\min_{1\le i<n}(p_{n+i}+p_{n-i})-2p_n .$$

By (1), the midpoint convex primes are exactly the $p_n$ with $M_n>0$. This is a definition and a restatement of (1); the paper proves no theorem about $M_n$.

Data (pp. 12--13). The paper tabulates the count $M(x)$ of midpoint convex primes up to $x$ for $x=10^1,\ldots,10^{11}$, with $M(10^{11})=1195764$ and $\log M(x)/\log x$ rising from $0.45154$ at $x=10^2$ to $0.55251$ at $x=10^{11}$. Figure 2 shows the distribution of $M_n$ for $n<1.6\times10^8$, and separately its nonnegative part. On this data the paper says that "it appears likely that $M_n$ can be arbitrarily large" (p. 13). It notes that $M_n$ can be arbitrarily negative, since $p_{n+1}+p_{n-1}-2p_n$ is a difference of consecutive prime gaps, and that the values of $M_n$ tend to avoid multiples of 6.

Questions (p. 14). Question 5.1 asks whether $M(x)=o(\pi(x))$, and likewise whether the count $G(x)$ of good primes is $o(\pi(x))$. Question 5.2 asks whether $C(x)=o(M(x))$ or $L(x)=o(G(x))$, where $C$ and $L$ count the convex and log-convex primes.

Read depth. Claims checked: equations (1) and (23), the table of $M(x)$, the caption of Figure 2, the sentence quoted above and Questions 5.1 and 5.2 were read on the page images of the preprint. The computations were not reproduced here.

Dependencies

None; the data are the author's computation.

Bears on

• Problem 454: $M_n$ is the problem's $f(n)-2p_n$ with the minimum taken over $1\le i<n$, so the problem asks whether $\limsup_n M_n=\infty$. The paper proves nothing about this. It presents histograms of $M_n$ for $n<1.6\times10^8$ and on them judges it likely that $M_n$ can be arbitrarily large: numerical evidence for a yes answer, not a proof.