Problem 246

Let $(a,b)=1$ . The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$ .

Second statement

Let $a,b\geq2$ with $(a,b)=1$ . The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$ .

Source: erdosproblems.com/246

Number theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The "(LEAN)" suffix is the site's catalog label; the theorem, its acceptance and the Lean development are recorded on the claim pages, and no Lean was built here.

Notes

The site's wording fixes only $(a,b)=1$ . Read as the site words it, it includes $a=1$ , where the set $\{b^l\}$ is complete only for $b=2$ ; for $b=3$ its subset sums, the integers with base-3 digits $0$ and $1$ , have density zero, and $a=b=1$ fails as well. The corrected Statement adds $a,b\geq2$ , the setting of Birch's theorem [Bi59], of the later literature and of the formal-conjectures statement; the standing judges it.

Bi59: Birch, B. J., Note on a problem of Erdős. Proc. Cambridge Philos. Soc. (1959), 370-373.
Ca60: Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.
Er61: Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
FaCh17: Fang, Jin-Hui and Chen, Yong-Gao, A quantitative form of the Erdős-Birch theorem. Acta Arith. (2017), 301-311.
He00b: Hegyvári, N., On the completeness of an exponential type sequence. Acta Math. Hungar. (2000), 127-135.
Yu24: Yu, Wang-Xing, On the representation of an exponential type sequence. Publ. Math. Debrecen (2024), 253-261.

Problem 246

Second statement

Status

Notes

Proof claims

Discussion