Is it possible to represent an improper fraction as a finite sum of unique unit fractions (Egyptian fractions)?
2026-02-23 05:33:35.1771824815
Represent improper fraction as a sum of unique unit fractions
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Yes. Start with $\alpha \in \mathbb Q$, $\alpha >1$. Then let $n$ be the greatest integer such that $$H_n=\sum_{i=1}^n\frac 1i<\alpha $$
Of course $n$ exists because the infinite Harmonic series diverges.
It follows that $\alpha - H_n<\frac 1n$ so none of the fractions in the standard Egyptian decomposition of $\alpha - H_n$ can appear in $H_n$.