Problem 1101

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claims/: The 1 claim page of Problem 1101, one per claimant's result; the problem's standing derives from them.

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Statement. If $u=\{u_1<u_2<\cdots\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\neq j$ and $\sum \frac{1}{u_i}<\infty$ then let $\{a_1<a_2<\cdots\}$ be the sequence of integers which are not divisible by any of the $u_i$. For any $x$ define $t_x$ by

$$u_1\cdots u_{t_x}\leq x< u_1\cdots u_{t_x}u_{t_x+1}.$$

We call such a sequence $u_i$ good if, for all $\epsilon>0$, if $x$ is sufficiently large then

$$\max_{a_k<x} (a_{k+1}-a_k) < (1+\epsilon)t_x \prod_{i}\left(1-\frac{1}{u_i}\right)^{-1}.$$

Is there a good sequence such that $u_n< n^{O(1)}$? Is there a good sequence such that $u_n\leq e^{o(n)}$?

Status. Open. The site labels the problem OPEN (page last edited 19 October 2025). Its two questions are the problem's two parts; Erdős expected the first to have a negative answer and the second a positive one. The second, whether some good sequence has $u_n\leq e^{o(n)}$, has one pending partial claim, Li's subexponential good sequence of 2026, which constructs a good sequence of primes with $u_n=\exp((1+o(1))\sqrt n)$. The first, whether some good sequence has $u_n<n^{O(1)}$, has no claim.

Source. erdosproblems.com/1101, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1101, https://www.erdosproblems.com/1101.

References.

• [Er81h] Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• erdos_1981_problems_results_additive_multiplicative_number_theory
• erdos_1981_problems_results_additive_multiplicative_number_theory / display_3_5