---

Claim. The largest integer that is not a sum of squares of distinct positive integers whose reciprocals sum to $1$ is $8542$; so every integer $m>8542$ is $n_1^2+\cdots+n_k^2$ with $1\le n_1<\cdots<n_k$ and $\sum1/n_i=1$, the case $p(x)=x^2$ of Problem 283 with the exact threshold. The theorem is Theorem 1 of M. A. Alekseyev, On partitions into squares of distinct integers whose reciprocals sum to 1, arXiv:1801.05928 (v1 18 January 2018, v2 23 April 2018), published as a chapter of The Mathematics of Various Entertaining Subjects, Volume 3 (J. Beineke and J. Rosenhouse, eds.), Princeton University Press, 2019, 213--221; the library's result page theorem_1 records the preprint's statement and the method: Graham's translations of representations of smaller numbers into representations of larger ones, a second class of translations on restricted representations, and an exhaustive search bounded by the power mean inequality that builds the starting representations and certifies that $8542$ has none. The site's commentary illustrates the case with $1=\frac12+\frac14+\frac16+\frac1{12}$ and $200=2^2+4^2+6^2+12^2$.

Covers. The polynomial $p(x)=x^2$ only, with $m>8542$ as the exact range. The theorem decides nothing for any other polynomial; the general case is the full claim Price 2026.

Depends on. Nothing in this wiki; the theorem and its computation are the paper's own.

Standing. Claimed. The chapter appeared in an edited volume, and neither the volume nor the library card records that it was refereed, so the page lists no refereed evidence. The site's commentary records that Alekseyev proved the case $p(x)=x^2$ for all $m>8542$, but the problem's label, PROVED (LEAN), settles the whole problem through the Price argument rather than this case, so the curator's credit is not reviewed evidence either. The corpus has checked the statement against the preprint, records the proof in outline only, has not rerun the computation, and has not compared the published chapter with the preprint.