---

Source. Published p. 590, Table 2 and the paragraph before it; the search method is on p. 589.

Statement. Table 2, headed as the smallest element of $S_n$ with its factorization, lists a value $k_n$ for each $1\le n\le20$, with $S_n$ the set defined on Theorem 2.1. For $n\le16$ the paper reports $k_n$ as the least element of $S_n$, computed with the criterion of Theorem 2.1; for $n=17$ a search over $k\le10^8$ found no element (p. 590). The rows $17\le n\le20$ carry an asterisk: the paper found these elements of $S_n$ through Conjecture 3.8 (p. 595), expects them to be the least, and did not check that. Row $n$ answers Problem 389 at $n+1$:

The starred values for $n=17$ and $n=19$ are $z_8-17$ and $z_9-19$ for the values $z_8$ and $z_9$ of Table 4, and those for $n=18$ and $n=20$ are one less, the pattern of Conjecture 2.11 ($k_{2n-1}-1\in S_{2n}$). The abstract (p. 583) states the outcome as a positive answer to Question 1.1 for each $n\le20$.

Proof pointer. A computer search (p. 589): for each candidate $k$ and each $j\le n$, the primes dividing $k+j$ are found, and the inequality of Theorem 2.1 is tested at each of them.

Dependencies. Theorem 2.1; for the starred rows, Table 4 and Conjecture 3.8 as the means of finding them.

Read depth. Claims checked: the table and the account of the search were read on the page images. No entry was recomputed and no search was rerun, so the minimality for $n\le16$ and the membership of the starred values rest on the paper's report. Nothing here is independently reviewed.

Bears on. #389: the table reports a witness $k$ for each of the problem's $n$ from $2$ to $21$, the least one for $n\le17$; at the problem's $n=1$ every $k$ works. These are instances only: the problem asks about every $n$.