conjecture on fractions with odd denominatorsAnyRationalNumberWithOddDenominatorIsASumOfUnitFractionsWithOddDenominators2002-06-23 06:42:582007-09-14 03:52:49Conjectureany rational number is a sum of unit fractions% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{conjecture}{Conjecture}Egyptian fractions raise many open problems; this is one of the most famous of them.
Suppose we wish to write fractions as sums of distinct unit fractions \emph{with odd denominators}. Obviously, every such sum will have a reduced representation with an odd denominator.
For instance, the greedy algorithm applied to $\frac{2}{7}$ gives $\frac{1}{4}+\frac{1}{28}$, but we may also write $\frac{2}{7}$ as $\frac{1}{7}+\frac{1}{9}+\frac{1}{35}+\frac{1}{315}$.
It is known that we can we represent \emph{every} rational number with odd denominator as a sum of distinct unit fractions with odd denominators.
However it is not known whether the \PMlinkname{greedy algorithm}{AnyRationalNumberIsASumOfUnitFractions} works when limited to odd denominators.
\begin{conjecture}
For any fraction $0\le \frac{a}{2k+1} < 1$ with odd denominator, if we repeatedly subtract the largest unit fraction with odd denominator that is smaller than our fraction, we will eventually reach 0.
\end{conjecture}