On a problem of Bleicher and Erdős
Let \(\mathbb Z_ 0\) be the set of positive integers and \(a,N\in\mathbb Z_ 0\) be such that \(a<N\). By an Egyptian fraction expansion of \(a/N\), we mean \(a/N=\sum_{i=1}^{k}1/n_ i\), \(n_ 1<n_ 2<...<n_ k\), where \(n_ i\in\mathbb Z_ 0\). Let \[ D(a,N)=\min \{n_ k: a/N=\sum_{i=1}^{k}1/n_ i,\quad n_ 1<n_ 2<\cdot \cdot \cdot <n_ k,\quad n_ i\in\mathbb Z_ 0\}, \] where the minimum ranges over all Egyptian fraction expansions of \(a/N\) and let \(D(N)=\max \{D(a,N): 1\leq a\leq N\}\). The author proves that \(D(N)/N\leq (\log N)^{1+\delta (N)},\) \(\delta(N) \to 0\) as \(N\to \infty\), establishing a conjecture of M. N. Bleicher and P. Erdős.
scientific article; zbMATH DE number 4063187