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Statement

Consider $\sum_{i=1}^n1/x_i=1$ with $2\times3\le x_1<x_2<\cdots<x_n$ and each $x_i=p_iq_i$ a product of two distinct primes (display (1), p. 1).

Main results (Section 2, p. 2). "We have the new 17 examples of (1) which have 47 terms." The seventeen representations are printed on pp. 5--10.

Section 4.1 (p. 11). "We proved that there are no 46-term examples when the maximum prime factor is 101." The largest prime factor among the denominators of the seventeen 47-term examples is $71$.

The paper adds (Section 4.1, p. 11) that, since the largest such prime factor is 71, another 47-term example beyond its seventeen is unlikely; this is the author's judgment, not a stated result.

Source. Tatsuru Watanabe, New examples of the representation of 1 by the sum of reciprocals of semiprime numbers, arXiv:2009.03275v2 (9 September 2020; the PDF is dated September 10, 2020), 11 pages: Section 2 on p. 2, Section 3 on pp. 3--10 with the examples in Section 3.3 on pp. 5--10, Section 4 on p. 11, all read on the printed pages. No journal version was found (arXiv listing and Crossref query of 2026-09-18). The edition is identified on the source card.

Read depth. Claims checked: the statement of Section 2, the history of Section 1.1 (Barbeau's 101-term solution of 1977, Johnson's 48-term solution of 1978, Guy's question whether $48$ is minimal) and Section 4 were read clause by clause; the seventeen displayed representations were counted on pp. 5--10 but not re-added here; the search (Sections 3.1--3.2, pp. 3--4, and the closing paragraph of Section 3.3, p. 5) was read for structure and not rerun.

Method (Section 3, pp. 3--10)

Johnson's denominators are factored (Section 3.1, p. 3); with the prime factors bounded by $53$ there are $120$ admissible semiprimes, and a $\{0,1\}$-vector in $\mathbb R^{120}$ records which reciprocals appear; a tree search guided by Proposition 1 (p. 4) finds two 47-term examples and $94$ examples with $48$ terms, Johnson's among them (Section 3.2, pp. 3--4); raising the prime bound to $101$ (a $325$-dimensional space, with what the paper calls minor revisions) gives fifteen more 47-term examples and no example with $46$ or fewer terms (Section 3.3, p. 5).

Dependencies

None beyond the computation, which is the author's and was not rerun here.

Bears on

• Problem 306: the instance $a/b=1$ only, which Barbeau (1977) and Johnson (1978) had already settled; the page gives representations of $1$ with two-prime denominators in $47$ terms, one fewer than Johnson's, and the bounded nonexistence of $46$-term ones; the least number of terms is not part of Problem 306 as stated, and the paper's expectation that $47$ is the minimum (abstract, p. 1: "it is assumed"; Section 4.2 lists a proof as future work) is not a theorem.