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Claim. If $n+1$ is divisible by a prime $p\equiv3\pmod4$, then $4/n$ is a sum of three unit fractions. The result is R. Obláth's, in Sur l'équation diophantienne $\frac4n=\frac1{x_1}+\frac1{x_2}+\frac1{x_3}$, Mathesis (4) 59 (1950), 308--316 (Zbl 0039.03403, a journal article); the record carries no day or month, so this page's date is the first of the year. The paper is not held and has not been read: the statement is second-hand, as Pomerance and Weingartner restate it ([PoWe25], pp. 1--2: "An early result of Obláth [11] is that $n$ has this property if $n+1$ is divisible by a prime $p\equiv3\pmod4$. This implies that asymptotically all $n$ have the Erdős--Straus property") and as Elsholtz and Tao record it ([ElTa13], p. 4).

As the corpus's own check, not as Obláth's text: if $n+1=(4t-1)b$ with $t,b\ge1$, then

$$\frac1{tb}+\frac1{ntb}+\frac1{nt}=\frac{n+1+b}{ntb}=\frac{4tb}{ntb}=\frac4n,$$

and a prime factor $p\equiv3\pmod4$ of $n+1$ gives $4t-1=p$.

Covers. Every $n>2$ such that $n+1$ has a prime factor congruent to $3$ modulo $4$, a set of density one among the integers, with the terms made distinct as the Formulation of Problem 242 records. Every other $n$ is not covered by this page.

Depends on. No page of this wiki.

Acceptance. Refereed: Mathesis 59 (1950), a journal. The site labels the problem FALSIFIABLE, an open problem, so its commentary's credit to Obláth is no reviewed evidence. The paper has not been read; the result is recorded from the restatements named above, and no proof review is supplied.