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Statement

Theorem 1.3. For some absolute constant $C>0$ the following holds. If $m$ and $N$ satisfy $4\le m\le\log^2N$, then at most

$$\frac{N}{\exp\bigl(C\log^{2/3}(N)/\varphi(m)^{1/3}\bigr)}$$

integers $n\le N$ have $m/n$ not expressible as a sum of $3$ unit fractions.

The paper adds: "Exploiting the large sieve, the proof is largely derivative of Vaughan's theorem in [13]" (p. 2), Vaughan's theorem being the bound $N/\exp(c\log^{2/3}N)$ for the exceptions to the Erdős--Straus conjecture (Mathematika 17 (1970), the paper's [13]; recalled on p. 2).

Source. Pomerance and Weingartner, arXiv:2511.16817v2 (15 January 2026), Theorem 1.3 on p. 2, read on the page image. Published as The Ramanujan Journal 69 (2026), no. 2, article 31, DOI 10.1007/s11139-025-01312-2; the published version was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page image; the proof was not read.

Dependencies

The large sieve, in the form of Vaughan's argument; not examined here.

Bears on

• Problem 242: with $m=4$ the theorem restates Vaughan's bound on the number of exceptions up to $N$, $N\exp(-c(\log N)^{2/3})$, from a held source (Vaughan's paper itself is not held); it says nothing about whether any exception exists.