<s>[INST] Egyptian fraction representation of $1$ where all denominators of the fractions are odd. Question: Is there an Egyptian fraction representation for $1$ where all the fractions have odd denominators?
I tried to generate one below:
$$\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\frac{1}{13}+\frac{1}{23}+\frac{1}{721}+\frac{109}{106711605}.$$
The last term can be further decomposed to: $$\frac{1}{979007}+\frac{158}{1.04471\cdot 10^{14}}.$$
or, it is impossible for any collection of $\frac{1}{n}$ where $n$ is odd to produce $1$?
 [/INST] $$\frac 13+\frac 15+\frac 17+\frac 19+\frac{1}{15}+\frac{1}{19}+\frac{1}{21}+\frac{1}{25}+\frac{1}{173}+\frac{1}{1294257}+\frac{1}{2233466948475}$$ is a solution
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