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Claim. There is a real number $M$ such that every rational number $m>M$ with more than three distinct representations $m=2^\alpha3^\beta+2^\gamma+3^\delta$, the exponents integers, has the form $m=2^a+3^b$, and for such $m$ the representations are the four given by

$$2^{a-1}3^0+2^{a-1}+3^b=2^{a-2}3^1+2^{a-2}+3^b =2^13^{b-1}+2^a+3^{b-1}=2^33^{b-2}+2^a+3^{b-2}.$$

Two representations count as distinct when their unordered triples of summands differ. This is Theorem 4 of R. Tijdeman and L. X. Wang, Sums of products of powers of given prime numbers, Pacific J. Math. 132 (1988), no. 1, 177--193, published 1988-03-01 by the publisher's record and received on 1986-10-24 by the paper's own dateline; a correction appeared in Pacific J. Math. 135 (1988), no. 2, 396--398 (it states that the paper's Lemma 3(b) is false and gives a corrected Lemma 3(b) and a new proof of Theorem 3 with the same solutions; no theorem statement changes, Theorem 4's included), which this corpus does not hold. Restricted to positive integers $n$ and nonnegative exponents, the theorem bounds the count $w(n)$ of Problem 407: each unordered triple of summands $\{2^a,3^b,2^c3^d\}$ arises from at most six ordered quadruples $(a,b,c,d)$, one per assignment of the three roles, so $w(n)\le24$ for $n>M$, and the finitely many $n\le M$ each have finitely many representations. The problem's question is therefore answered affirmatively a second time, with the best possible eventual bound of four distinct representations, which the earlier proof of [[problems/diophantine_problems/E0407/claims/1988_10_13_evertse_gyory_stewart_tijdeman|Evertse, Győry, Stewart and Tijdeman]] does not give; that proof is earlier although its Durham 1986 chapter was printed in October 1988, after this paper, which already cites it as having settled the conjecture. The constant $M$ is ineffective, so this theorem, like the earlier proof, gives no computable bound on $w(n)$ for every $n$ (Bajpai and Bennett note that no explicit threshold can be extracted from its argument): the proof uses the finiteness theorem for $S$-unit equations (the paper's Lemma 4, after van der Poorten and Schlickewei and Evertse) together with the complete solution, by Baker's method, of the three exponential equations $2^x3^y+1=2^z+3^w$, $2^x3^y+2^z=3^w+1$ and $2^x3^y+3^w=2^z+1$ (the paper's Theorems 1--3). The site's commentary records the result as $w(n)\le4$ for all large $n$ under the distinct-summand convention. This page rests on the statement of Theorem 4 and the introduction of the paper (card); the proof was not checked.

Acceptance. The paper is a refereed publication in the Pacific Journal of Mathematics, 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 quantitative bound; that documented acceptance is the reviewed evidence. Bajpai and Bennett's introduction restates the theorem as their Theorem 2 and builds on it; their effective version has its own page.