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Claim. Let $\omega(N)$ count the nonnegative integer tuples $(a,b,c,d)$ with $N=2^a3^b+2^c+3^d$, two tuples identified when their summand sets $\{2^a3^b,2^c,3^d\}$ agree. Then $\omega(N)\le9$ for every positive integer $N$, and $\omega(N)\le8,7,6,5,4$ for $N\ge300,786,2316,19700,131082$ respectively; $\omega(N)=9$ exactly for $N\in\{41,83,89,113,137,161,227,299\}$, the largest $N$ with $\omega(N)=5,6,7,8$ are $131081,19699,2315,785$, and $\omega(N)=4$ for infinitely many $N$, by the identities for $N=2^a+3^b$. This is Theorem 3 of P. Bajpai and M. A. Bennett, Effective $S$-unit equations beyond three terms: Newman's conjecture, Acta Arith. 214 (2024), 421--458, first posted as arXiv:2308.05162 on 2023-08-09. Under the problem's own convention, which counts ordered quadruples, each summand set arises from at most six quadruples, so the theorem gives an explicit bound on $w(n)$ for every $n$ and answers the question of Problem 407 affirmatively with computable constants, where the earlier proofs of Evertse, Győry, Stewart and Tijdeman and of Tijdeman and Wang were ineffective. The site's commentary states the bounds as $w(n)\le4$ for $n\ge131082$ and $w(n)\le9$ for all $n$, with the largest $n$ of count nine being $299$. The proof rests on the paper's Theorem 1, an effective bound for the heights of nondegenerate solutions of five-term $S$-unit equations over a number field when $S$ has at most three places, obtained from lower bounds for linear forms in complex and $p$-adic logarithms and a matching procedure that reduces a five-term equation to the four-term case; the corpus's card summarizes it. This page rests on the statement of Theorem 3 and the introduction of the paper; the proof was not checked.

Acceptance. The paper is a refereed publication in Acta Arithmetica, the refereed evidence. The site's curator, T. F. Bloom, labels the problem proved and credits this paper in the problem's commentary with the effective bounds, noting in the site's thread on 2025-09-08 that the details and references had been added; that documented acceptance is the reviewed evidence. The thread also carries a reader's computed values under the ordered convention, which the thread itself attributes to double counting of permuted summands; those posts are unverified comments and change nothing here. The Lean development recorded on the page of Evertse, Győry, Stewart and Tijdeman names this paper among its informal sources and proves, as a conditional theorem, that the ordered count is at most $27$ if the bound nine holds; the nine bound itself is a hypothesis there, not a formalized result.