Problem 845

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claims/: The 1 claim page of Problem 845, one per claimant's result; the problem's standing derives from them.

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Statement. Let $C>0$. Is it true that the set of integers of the form

$$n=b_1+\cdots+b_t\textrm{ with }b_1<\cdots<b_t$$

where $b_i=2^{k_i}3^{l_i}$ for $1\leq i\leq t$ and $b_t\leq Cb_1$ has density $0$?

Formulation. The question is read for every $C>0$: as Erdős posed it ([Er92b], Problem 21, p. 239: for summands $2^k3^l$ almost all integers should fail even when $b_t/b_1$ may be a large constant), as the site labels it, and as the formal-conjectures statement quantifies it. The standing concerns that reading. For a single $C$ the answer is yes for $C<3$ (density zero), no for $C\ge6$ (every positive integer is such a sum), and unproved for $3\le C<6$; see the van Doorn-Everts claim.

Status. DISPROVED (LEAN).

Source. erdosproblems.com/845, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #845, https://www.erdosproblems.com/845.

References.

• [Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
• [ErLe96] Erdős, P. and Lewin, Mordechai, [[../library/diophantine_problems/erdos_1996_d_complete_sequences_integers/_index|$d$-complete sequences of integers]]. Math. Comp. (1996), 837-840.
• [vDEv25] W. van Doorn and A. Everts, Smooth sums with small spacings. arXiv:2511.04585 (2025).

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• doorn_2025_smooth_sums_small_spacings
• doorn_2025_smooth_sums_small_spacings / lemma_1
• doorn_2025_smooth_sums_small_spacings / theorem
• erdos_1996_d_complete_sequences_integers
• erdos_1992_my_favourite_problems_various_branches_combinatorics