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Claim. Steve Fan, Strongly complete sets and a conjecture of Erdős, arXiv:2607.14071; Theorem 1.5 first appears in the fourth version, of 9 September 2026, and the fifth version is of 16 September 2026 (the source card is fan_2026_strongly_complete_sets_conjecture_erdos, which records the theorem at statement level). Let $D$ be a finite nonempty set of integers at least $2$ in which no two elements are powers of one integer, partitioned as $D=D_1\cup D_2\cup C$ with $\gcd(C)=1$ and $\sum_{d\in D_i}1/(d-1)\ge1$ for $i=1,2$. Then the set of all powers $d^n$ with $d\in D$ and $n\ge0$ is strongly complete: it stays complete, every sufficiently large integer being a sum of distinct elements, after any finite subset is removed. The theorem refines Theorem 1.23 of Bergelson and Simmons (claim page), which needs three parts with reciprocal sums at least $1$ beside the gcd part. The paper's AI disclosure, as the source card records it, says that ChatGPT 5.6 was used for proofreading and suggested a core idea of the shorter proof of its Lemma 3.2. The argument was not reconstructed in this corpus.

Covers. The second question of Problem 124, for every $k\ge1$, at every tuple $d_1<\cdots<d_r$ that admits such a partition: yes, since the powers with exponent below $k$ form a finite set and a sum of distinct powers $d^n$ with $n\ge k$ is $\sum_ic_ia_i$ with $a_i\in P(d_i,k)$. Not covered: tuples without such a partition, in particular every tuple whose reciprocal sum is at most $2$, and the first question.

Depends on. No page of this wiki.

Standing. Claimed. The paper is an arXiv preprint with no journal record and no published independent review. The site's commentary (page last edited 1 December 2025) predates the theorem and labels the problem OPEN; a post on the site's thread of 23 September 2026 cites the theorem as settling the second question when the reciprocal sum exceeds $2$. No Lean formalization of the theorem is known, and this corpus has built nothing, so the page lists no evidence.