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Claim. [[../library/diophantine_problems/borwein_1990_questions_erdos_graham_numbers_form_sum_g_n_2_g_n/corollary_1|Corollary 1]] (p. 381) of P. B. Borwein and T. A. Loring, Some questions of Erdős and Graham on numbers of the form $\sum g_n/2^{g_n}$, Math. Comp. 54 (1990), no. 189, 377--394 (library card): if the paper's Conjecture 1 holds, then every dyadic rational has a terminating $*$-binary representation $\alpha=\sum_{n\ge1}n\,d_n/2^n$ with digits $d_n\in\{0,1\}$. The paper does not state the consequence for the second question of Problem 261 as such; it follows from the paper's splitting (2.5) of $(m-1)/2^{m-1}$ as $m/2^m$ plus a sum of distinct terms $k/2^k$ with $k>m$, which the paper notes (p. 384) is finite under Conjecture 1. So under the conjecture $n/2^n$ is a sum of at least two distinct terms $k/2^k$ for every $n\ge2$, and the case $n=1$ holds unconditionally; the corollary's result page records both. The representation comes from the paper's greedy algorithm (Algorithm 1), which writes $\alpha=\sum b_n/2^n$ in binary, sets $a_1=b_1$ and $a_{n+1}=2(a_n\bmod n)+b_{n+1}$, and puts $d_n=1$ exactly when $a_n\ge n$ (pp. 379 and 380 print the update with $+b_n$, a misprint for the $+b_{n+1}$ of the paper's (2.2)).

Hypothesis. [[../library/diophantine_problems/borwein_1990_questions_erdos_graham_numbers_form_sum_g_n_2_g_n/conjecture_1|Conjecture 1]] (p. 379): for any integer starting value $a_m$, the iteration $a_{n+1}=2(a_n\bmod n)$ eventually reaches $0$; the print does not specify the range of the starting index $m$. The paper supports it by computation (Proposition 8: for base $2$ the iteration terminates for every positive initial value when $m\le1000$, and Section 5 tabulates the termination function) and does not prove it. The hypothesis is unproved, and the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself. Unconditionally, the second question is verified for $n\le10^4$ on Tengely, Ulas and Zygadło's page and is otherwise open; the first question is answered on the paper's unconditional page.

Depends on. Nothing in this wiki; the hypothesis is stated above.

Acceptance. Refereed: Mathematics of Computation 54 (1990), no. 189, 377--394 (refereed). The site labels the problem OPEN, so no curator acceptance is listed. The library holds no file of the paper; the statement is recorded from its card, and the corpus records no check of the proof.

Dating. The page is dated by the issue month in the publisher's record, January 1990; the day is a placeholder.