Let $$\epsilon_m=\max \sum \frac{1}{n_i}$$ where the maximum is taken over all finite sequences $m<n_1<\cdots<n_k$ for which there exist congruences $a_i\pmod{n_i}$ such that no integer satisfies two such congruences.

Let $$\epsilon_m=\max \sum \frac{1}{n_i}$$ where the maximum is taken over all finite sequences $m<n_1<\cdots<n_k$ for which there exist congruences $a_i\pmod{n_i}$ such that no integer satisfies two such congruences.

Estimate $\epsilon_m$.

The resolution of [202] by GPT-5.4 Pro implies via the same reduction that $$\epsilon_m=L(m)^{-1+o(1)},$$ where $L(m)=\exp(\sqrt{\log m\log\log m})$.