Problem 345

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Statement. Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq m$ are in

$$P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$$

(the existence of $T(A)$ is guaranteed by completeness).

Is it true that there are infinitely many $k$ such that $T(n^k)>T(n^{k+1})$?

Status. Open.

Source. erdosproblems.com/345, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #345, https://www.erdosproblems.com/345.

References.

• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Formalization. None recorded.

Progress

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Known Results

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