Problem #453: Is it true that, for all sufficiently large $n$, there...

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Progress

Asked by Erdős and Straus. Selfridge conjectured the answer is no, contrary to Erdős and Straus. The answer is no, as shown by Pomerance [Po79], who proved there are infinitely many $n$ such that\[p_n^2 > p_{n+i}p_{n-i}\]for all $i<n$. Pomerance's proof is short enough to reproduce here:

Let $0<a_1<a_2<\cdots$ be any sequence with $a_n=o(n)$. Consider the boundary of the convex hull of the points $(n,a_n)$. The fact that $a_n/n\to 0$ implies that the non-horizontal portion of the boundary of the convex hull is concave and has infinitely many vertices. Considering these vertices as $(n,a_n)$ produces infinitely many $n$ such that $2a_n > a_{n-i}+a_{n+i}$ for all $i<n$. We may then choose $a_n=\log p_n$.

This is the topic of problem A14 of Guy's collection [Gu04].