Two corrections to §8 of Shiu (2016): E_2113 contains 1364, and the last run at 10^4 has 156 members
Year: 2026
Type: article
Abstract: Section 8 of J. Shiu, "Denominators of harmonic numbers" (arXiv:1607.02863v2), contains two typographical errors. (i) Table 2 lists E_2113 = {44, 443, 553, 748, 1384, 1559, 1669, 2068, 2112}. Shiu himself proves that E_p is invariant under the involution m -> p-1-m, for 1 <= m <= p-2. For p = 2113 this forces 1364 (since 748+1364+1 = 2113), whereas the printed 1384 would force 728. Direct enumeration confirms 1364 is in E_2113 and 728 is not. The entry 1384 should read 1364; the cardinality |E_2113| = 9 is unchanged. (ii) The last of the 26 maximal runs of good n up to 10^4 is printed as 9156_155; the printed run lengths sum to 2640, whereas the text states that there are 2641 such n. Exact computation gives 26 runs summing to 2641, the last being 9156 <= n <= 9311 with 156 members. This second point has already been recorded in the public research notes at erdosproblems.ai (problem 291); it is included here for completeness and no priority is claimed for it. Reproducible code (Python standard library only) is contained in the attached document.
Author: Tian, Yuanye
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225737324
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