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 $1$?
Best Answer
The answer is yes and I have shown how, but I can't find the previous question. It takes at least $9$ fractions. This page has many expansions. The one with the smallest maximum denominator is $$1=\frac 13+\frac 1 5+\frac 1 7+\frac 1 9+\frac 1 {11}+\frac 1{ 15}+\frac 1 {35}+\frac 1{45}+\frac 1{ 231}$$