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Source. Conjecture 3.1 (p. 8) and Theorems 3.2 and 3.3 (p. 9), Section 3, 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

An edge convex prime is a prime $p_n$ whose point $(n,p_n)$ lies on the boundary of the convex hull of the prime number graph without being a vertex of it; for example $(3,5)$ lies on the segment from $(2,3)$ to $(4,7)$ (p. 8). The counts below are of edge convex primes up to $x$.

Theorem 3.2 (p. 9). For some constant $b'>0$ the number of edge convex primes up to $x$ is

$$O\Bigl(x\exp\Bigl\{-b'\frac{\log^{3/5}x}{(\log\log x)^{1/5}}\Bigr\}\Bigr).$$

Theorem 3.3 (p. 9). Assuming the Riemann Hypothesis, the number of edge convex primes up to $x$ is $O(x^{7/8}\log^{3/4}x)$.

Conjecture 3.1 (p. 8). There are only finitely many edge convex primes. The computation to $10^{13}$ found exactly five, namely $5,13,23,31,43$ (pp. 8, 12).

Read depth. Claims checked: Conjecture 3.1 and Theorems 3.2 and 3.3 were read clause by clause on the page images of the preprint. The proof of Theorem 3.2 was read but not checked. Theorem 3.3 is stated without proof, as the improvement that Theorem 2.4 gives (p. 9). Nothing here is independently reviewed.

Proof pointer

p. 9. Count in $(\tfrac12x,x]$. By Theorem 2.3 each boundary segment spans $O(x\exp\{-B\log^{3/5}x/(\log\log x)^{1/5}\})$ primes. On a segment of slope $a/d$ in lowest terms the edge convex primes are at least $d$ primes apart. Since the slopes lie in an interval of length $\log2+o(1)$ (Lemma 2.1, p. 4), there are $O(\varphi(d))$ segments whose slope has denominator $d$. Splitting at a denominator bound $D$ and optimizing $D$ gives $O(x\exp\{-\tfrac12B\log^{3/5}x/(\log\log x)^{1/5}\})$ for the dyadic block; then sum dyadically.

Dependencies

Lemma 2.1, Theorem 2.3, and for Theorem 3.3 Theorem 2.4.

Bears on

No Erdős problem directly.