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Claim. For every sufficiently large $k$ there are integers $n_1<\cdots<n_k$ in an interval of width $(e-1+o(1))k$ with $1=1/n_1+\cdots+1/n_k$, which answers the question of Problem 286 as the site states it. The source is Martin's Theorem 2 (Denser Egyptian fractions, Acta Arith. 95 (2000), no. 3, 231--260; arXiv:math/9811112, 18 November 1998): for every positive rational $r$ and all $t\ge t_0(r)$ the least largest denominator in a $t$-term representation of $r$ by distinct unit fractions is $M_t(r)=t/(1-e^{-r})+O_r(t\log\log3t/\log3t)$. At $r=1$ this gives, for every $k\ge3$, a $k$-term representation of $1$ whose denominators all lie in $[2,M_k(1)]$ with $M_k(1)=(e/(e-1)+o(1))k$. Since $e/(e-1)\approx1.582<e-1\approx1.718$, the interval $[2,M_k(1)]$ has width below $(e-1)k$ for every large $k$, so the required interval exists with $o(1)=0$. This one-line deduction is the corpus's own; Martin's paper states Theorem 2 as the resolution of Problem 285. It discusses the width only as the least width $M'_t(r)$ of a $t$-term representation (pp. 2--3): Croot's result gives $M'_t(r)=t+O_r(t\log\log t/\log t)$ for infinitely many $t$, and no analogue of Theorem 2 valid for all $t$ is obtained for it. The paper does not draw the deduction above.

Readings. The site's question asks for an interval of width $(e-1+o(1))k$ containing the denominators, and the deduction answers it. The 1980 monograph, p. 33, printed the question as an equality for the least width, $\min\{x_n-x_1\}=(e-1)n+o(n)$; under that reading the same bound shows that the least width is at most $M_k(1)-2=(e/(e-1)+o(1))k$, below $(e-1)k$, so the printed equality is false for large $k$, as Croot's introduction notes, and Croot's own form of the question asks whether the least width is $\sim k$. The problem page's Formulation records the difference; this page answers the site's wording.

Depends on. Martin 1998, the accepted claim on Problem 285, which records Theorem 2 and its acceptance.

Acceptance. The paper is published in Acta Arithmetica, a refereed journal (Crossref record of DOI 10.4064/aa-95-3-231-260), which is the refereed evidence. The site's curator credits the problem to Croot, not to Martin, so no reviewed evidence is listed for this page. Proof coverage: as on the Problem 285 claim page, the reduction of Theorem 2 to Propositions 5 and 6 is recorded on the theorem page, and the proofs of the propositions have not been checked.

Formalization. The formalization link is a public Lean 4 proof of the every-large-$k$ statement in Boris Alexeev's lean-proofs repository (file of 2026-08-17, pinned to the commit of 2026-09-15), whose header names Croot and Martin as the informal authors and Codex and GPT-5.6 Sol as the formal authors. Its theorem erdos_286 takes the route above: it imports the repository's formalization of Martin's upper bound and uses the inequality $e/(e-1)<e-1$ (densityConstant_lt_exp_sub_one) to place the denominators in $[1,1+(e-1)k]$, with the error function identically zero; the file contains no sorry. formal-conjectures added ErdosProblems/286.lean on 2026-09-20 with this file as its formal_proof, and the community database records the problem as formalized since that date. The corpus has not built or audited the development, so the claim lists no formalized evidence.

Related. Croot's Main Theorem gives the sharper width $(1+o(1))k$ for infinitely many $k$, the accepted partial claim Croot 1999; the companion smallest-denominator question is Problem 284.