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Claim. Every prime $n$ coprime to $120120$ that is not congruent modulo $120120$ to one of the $198$ residues of the paper's Table 2 satisfies Rosati's condition (2) or (3), so that $4/n=1/x+1/y+1/z$ has a solution in positive integers. This is the outcome of the first algorithm of D. G. Terzi, On a conjecture by Erdös-Straus, BIT 11 (1971), 212--216, received 23 September 1970, the date this page carries: the abstract (p. 212) states it for primes, and congruence (9) with Table 2 (p. 214) lists the $198$ classes left open. The paper allows repeated terms; the Formulation of Problem 242 records how a representation becomes one with three distinct terms.

Covers. The primes coprime to $120120$ outside the $198$ classes, and, since a representation for a prime scales to one for every multiple of it, every $n>2$ with such a prime factor. The page does not cover a composite $n$ outside the $198$ classes whose prime factors all lie inside them: the classes are not closed under multiplication, for example $66049\cdot30241\equiv32449\pmod{120120}$ with $66049$ and $30241$ in Table 2 and $32449$ not. The site's commentary, which credits Terzi with every $n$ outside the classes, therefore says more than the paper's statement for primes. Nor does the page cover the paper's verification of all $n\le10^8$ (p. 215), an author's report of a computation that [ElTa13] finds incomplete and that the problem page records under Finite verification.

Depends on. No page of this wiki.

Acceptance. Refereed: BIT 11 (1971). The site labels the problem FALSIFIABLE, an open problem, so its commentary's credit to Terzi is no reviewed evidence. The statement and Table 2 are recorded on the library's result page; the algorithm is recorded in outline only, and the proof is not independently reviewed.