Problem 123

Let $a,b,c> 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ( $k,l,m\geq 0$ ), none of which divide any other?

Source: erdosproblems.com/123

Number theory

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SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN); the proof and its acceptance are recorded on the claim pages, and this corpus has built none of the Lean.

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.
ChYu23b: Chen, Yong-Gao and Yu, Wang-Xing, On $d$ -complete sequences of integers, II. Acta Arith. (2023), 161-181.
Er92b: Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
Er97: Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
Er97e: Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
ErLe96: Erdős, P. and Lewin, Mordechai, $d$ -complete sequences of integers. Math. Comp. (1996), 837-840.
MaCh16: Ma, Mi-Mi and Chen, Yong-Gao, On $d$ -complete sequences of integers. J. Number Theory (2016), 1-12.

Problem 123

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