Length and denominators of Egyptian fractions (Q1081626)

Let \[ L(a,N)=\min \{k: a/N=\sum^{k}_{1}1/n_ i,\quad n_ i\in {\mathbb{Z}}_ 0\}, \] \[ D(a,N)=\min \quad \max \{n_ k: a/N=\sum^{k}_{1}1/n_ i,\quad n_ i\in {\mathbb{Z}}_ 0\}, \] where \({\mathbb{Z}}_ 0\) is the set of positive integers and \(a,N\in {\mathbb{Z}}_ 0\), \(a<N\). Define \[ L(N)=\max \{L(a,N): 1\leq a<N\},\quad D(N)=\max \{D(a,N): 1\leq a<N\}. \] The author proves (1) that there is no algorithm which yields both \[ L(P)\leq c \log P/\log \log P\quad and\quad D(P)\leq P(\log P)^{1+1/(c+\epsilon)}\quad for\quad \epsilon >0; \] \[ (2)\quad L(N)\leq \frac{4 \log N}{\log \log N}(1+\frac{\log \log \log N}{\log \log N}) \] and D(N)\(\leq \lambda N(\log N)^ 2\), where \(\lambda\to 1\) as \(N\to \infty\).