Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 261

claims/: The 3 claim pages of Problem 261, one per claimant's result; the problem's standing derives from them.

Statement. Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that

\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?

Is this true for all $n$ ? Is there a rational $x$ such that

x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}

has at least $2^{\aleph_0}$ solutions?

Status. Open. The site labels the problem OPEN (page last edited 1 December 2025; accessed 2026-10-07). Its remarks credit Borwein and Loring [BoLo90] with an identity giving infinitely many $n$ , which answers the first question yes, and Tengely, Ulas and Zygadło [TUZ20] with a check of the second question for every $n\le10000$ ; both are refereed results with accepted partial claim pages, Borwein and Loring 1990 and Tengely, Ulas and Zygadło 2020, and Borwein and Loring's reduction of the second question to a termination conjecture is a conditional claim page. No claim settles or pends on the second or the third question, so the derived standing is open, claim none. The three questions are the parts infinitely_many, all_n and continuum of the frontmatter.

Source. erdosproblems.com/261, accessed 2026-09-04 and 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #261, https://www.erdosproblems.com/261.

References.

[BoLo90] Borwein, Peter and Loring, Terry A., [[../library/diophantine_problems/borwein_1990_questions_erdos_graham_numbers_form_sum_g_n_2_g_n/_index|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.
[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.
[TUZ20] Tengely, Szabolcs and Ulas, Maciej and Zygadło, Jakub, On a Diophantine equation of Erdős and Graham. J. Number Theory 217 (2020), 445-459.

Formalization. Statement in formal-conjectures. The file states the three questions as erdos_261.parts.i (tagged research solved), erdos_261.parts.ii and erdos_261.parts.iii (both research open), all with proof sorry; its variants state the Borwein--Loring identity and the property it gives (textbook), the check for $n\le10000$ (research solved, sorry) and Erdős's weakened two-representation question (research solved, with a formal_proof pointer to the outside Lean file recorded under Progress and known results). The community database records the statement as formalized since 2 September 2026 and no formal proof. The corpus has not built any of these files.

Current assessment

The question (site formulation as accessed 2026-10-07). The three questions above; status OPEN, last edited 1 December 2025; source keys [Er74b], [ErGr80] and [Er88c, p. 104]; the site relates the problem to Problem 260. The remarks record Cusick's unpublished proof for the first question, Borwein and Loring's identity, the check of every $n\le10000$ by Tengely, Ulas and Zygadło, and Erdős's weakening of the third question to a rational with two representations. The thread has three comments (14 August 2025 to 5 May 2026) and the proof-claim tab is empty.

Origin. Erdős and Graham [ErGr80] ask whether $n/2^n$ is a sum of at least two distinct terms $a_k/2^{a_k}$ for infinitely many $n$ , whether it is for every $n$ , and whether some rational has $2^{\aleph_0}$ such infinite representations; Erdős [Er88c, p. 104] repeats the questions, records that Cusick communicated a simple proof of the first to him in June 1987 without giving it, and asks only for a rational with two representations.

What is proved. The first question is answered yes: Borwein and Loring's Proposition 1 gives, for every positive integer $m$ and $n=2^{m+1}-m-2$ , $n/2^n=\sum_{n<k\le n+m}k/2^k$ , on their claim page. The second question is open: Tengely, Ulas and Zygadło verify it for every $n\le10^4$ (their claim page), and show that for each fixed number $k$ of terms there are only finitely many effectively computable solutions; Borwein and Loring's Corollary 1 reduces the question to their Conjecture 1, that the iteration $a\mapsto2(a\bmod n)$ always reaches $0$ (conditional claim page). The third question is open: Borwein and Loring give a dense set of irrationals with uncountably many representations (Proposition 3), under Conjecture 1 infinitely many terminating representations of every dyadic rational, and rationals with a unique representation (Proposition 5); Tengely, Ulas and Zygadło give infinitely many rationals with at least nine representations; none decides whether some rational has $2^{\aleph_0}$ .

The two-representation variant. Erdős's weakened question is a variant and settles no part of the problem. A thread comment of 27 April 2026 (Zeraoulia Rafik) answers it: since $4/2^4=5/2^5+6/2^6$ and $\sum_{m\ge1}m/2^m=2$ , the rational $7/4$ is represented both over $\mathbb N\setminus\{4\}$ and over $\mathbb N\setminus\{5,6\}$ ; a comment of 5 May 2026 (Vjekoslav Kovač) says the word two is generally believed to be a misprint for $2^{\aleph_0}$ . formal-conjectures marks its variant erdos_261.variants.two_representations research solved and cites a Lean proof of it that follows the comment; the corpus has not built it. A thread comment is not a dated manuscript and the variant is not the question, so neither has a claim page.

Search scope (2026-10-07 UTC). The site's problem page, its discussion thread and its proof-claim tab; the community database record; the formal-conjectures file at the revision linked above; the Crossref records of [BoLo90] and [TUZ20] and the arXiv listing of 2008.01501; the library cards of [BoLo90] and [TUZ20]. Not searched: MathSciNet, zbMATH, Google Scholar, X.

Proof coverage. The library holds no file of [BoLo90], [Er88c] or [TUZ20]; the results are recorded from the library cards and the site, and no proof or computation is checked in this corpus.

Progress and known results

Borwein and Loring (1990) answer the first question with an explicit identity, reduce the second to a termination conjecture and study the multiplicity of representations; Tengely, Ulas and Zygadło (2020) verify the second question for $n\le10^4$ , bound the solutions for each fixed number of terms and enumerate them for at most eight terms. The claim pages above record what each settles.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (24)

Borwein and Loring: infinitely many n with n over 2^n a sum of k over 2^k Borwein and Loring: every n has the property if their iteration always terminates Tengely, Ulas and Zygadło: every n up to 10000 has the property problems/diophantine_problems/E0261/claims Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/borwein_1990_questions_erdos_graham_numbers_form_sum_g_n_2_g_n Algorithm 1: the greedy algorithm as a base change from binary to *-binary digits Conjecture 1: the iteration a_(n+1) = 2(a_n mod n) always reaches zero Corollary 1: under Conjecture 1 every dyadic rational has a terminating *-binary representation Proposition 1: (m-1)/2^(m-1) is the sum of k/2^k over m <= k <= m+M-2 when m = 2^M - M Proposition 3: dense irrationals with uncountably many *-binary representations, and multiplicity results Proposition 5: the tails of sums of 2k/2^(2k) and (2k-1)/2^(2k-1) have a unique *-binary representation Proposition 8: the iteration a_(n+1) = c(a_n mod n) terminates for small starting indices diophantine_problems/tengely_2020_diophantine_equation_erdos_graham Conjecture 3.7 (p. 11): every solution has n + k <= a_k <= 2(n + k)Corollary 2.9 (p. 6): a_k <= 2^(k+2) + 2k(log_2 k - 1) - 4, so finitely many solutions for each k Corollary 3.6 (p. 10): infinitely many rationals with at least nine representations Proposition 3.1 (p. 6): an explicit residue class of k with at least five solutions Theorem 2.1 (p. 2): necessary conditions on a solution of n/2^n = sum a_i/2^(a_i)Theorem 2.5 (p. 5): all solutions of n/2^n = sum a_i/2^(a_i) with 2 <= k <= 8 terms Theorem 2.8 (p. 6): the largest term of a solution is at most 2n + 2k log_2 k Theorem 3.5 (p. 8): every n with 2 <= n <= 10^4 has a solution with a_1 = n + 1 irrationality/erdos_1988_irrationality_certain_series_problems_results

Graph