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Claim. Van Doorn and Everts prove that every positive integer $n$ is a sum $n=b_1+\cdots+b_r$ of distinct integers $b_i=2^{k_i}3^{l_i}$ with $b_1<\cdots<b_r<6b_1$. For $C=6$ the set that Problem 845 asks about is therefore the set of all positive integers, of density one, and the answer to the question is no. The result is the case $p=3$ of their main theorem: for every odd $p>1$ there is a constant $C_p$ such that every positive integer is a sum of distinct integers $2^xp^y$ whose largest term is below $C_p$ times the smallest, with $C_p=2p$ when $p-1$ is a power of two, $C_p=2(p+1)$ when $p+1$ is, and $C_p=\tfrac12F(4p)$ in general for an iterated-logarithm product $F$ defined in the paper. The construction adapts the explicit procedure of Blecksmith, McCallum and Selfridge for $d$-complete sets of 3-smooth numbers. The library card is van Doorn and Everts 2025.

Which constants. The same theorem shows that no constant below $p$ works: for $C<3$ the sums in question are too few, so the set has density zero and the question's answer for such $C$ is yes. Erdős and Lewin had shown that $C\le2$ fails (Erdős and Lewin 1996, p. 838). Between $3$ and $6$ the answer is not proved. In the site's thread, Cambie checked that $C=32/9$ works for every $n\le10^5$, the value the paper cites; Alexeev then found the first failure at $n=353515$ and checked that $C=3^{10}/2^{14}\approx3.604$ works, with $b_r\le Cb_1$, for every $n\le10^9$, with $167$ numbers up to $10^6$ needing exactly that value; Alexeev wrote that they think this value is optimal and, after checking over a hundred further members of a candidate extremal sequence, that they are no longer sure it is attained infinitely often. The original conjecture of Erdős 1992, Problem 21 (p. 239), expected that almost all integers fail to be such a sum for any constant; the site reads the question as that conjecture and labels it disproved, which is the standing recorded here.

Acceptance. Van Doorn announced in the site's thread on 2025-10-23 that van Doorn and Everts could resolve the problem in the negative, and posted the arXiv paper there on 2025-11-07. The site's curator, Thomas Bloom, credits the disproof to van Doorn and Everts with $C=6$ and labels the problem DISPROVED (LEAN), which is the reviewed evidence; the community database records the problem as disproved (Lean). The paper is an arXiv preprint: its arXiv record lists no journal reference and no published version is known, so there is no refereed evidence.

Formalizations. Two Lean developments are on record, neither built or audited by this corpus, so neither is formalized evidence. Boris Alexeev reported in the thread on 2026-01-08 a formalization produced by Aristotle (Harmonic) from the paper, retained in Alexeev's lean-proofs repository at the pinned commit above (Lean 4.24.0, Mathlib v4.24.0); its header names van Doorn and Everts as the authors of the proof, and it proves the main theorem for every odd $p$, the existence of some constant $C$ for the 3-smooth case, and the formal-conjectures statement of the problem with the answer false; the header says the $C=6$ bound itself was not formalized. Wouter van Doorn then rewrote the argument for $p=3$ alone and produced a Lean proof of the $C=6$ statement with Aristotle, announced in the thread on 2026-01-21 and retained in van Doorn's Lean-files repository at the pinned commit above (committed 2026-03-02, Lean 4.24.0); the file's header credits Aristotle (Harmonic) and names ChatGPT, Google, Gemini and Claude as the other systems used. The formal-conjectures statement file points to Alexeev's file.