Problem 125

Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$ , and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $4$ .

Does $A+B$ have positive lower density?

Source: erdosproblems.com/125

Number theory
Base representations

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The site labels the problem DISPROVED (LEAN). No manuscript or refereed publication of the disproof exists. Its written forms are Lean files, none built or audited in this corpus (the collection's pinned file, a community file and a generated file); the claimant's summary of the argument on the site's thread (post 5110); and the reconstructions by the site's curator (post 5114), who credits the result to DeepMind, and by Nat Sothanaphan (post 5115, crediting GPT-5.4 Thinking). Terence Tao's reconstruction (post 4469) covers only the earlier positive-density result. See "Current assessment" and the claim pages.

BEGL96: Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. 77 (1996), 133-138.
Er97: Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
HaMe24: M. Hasler and G. Melfi, On sums of distinct powers of $3$ and $4$ . Combinatorics and Number Theory (2024).
Me01: Melfi, Giuseppe, An additive problem about powers of fixed integers. Rend. Circ. Mat. Palermo (2) (2001), 239-246.

Problem 125

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Proof claims

Discussion