conjecture on fractions with odd denominators

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 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 every rational number with odd denominator as a sum of distinct unit fractions with odd denominators.

However it is not known whether the greedy algorithm (http://planetmath.org/AnyRationalNumberIsASumOfUnitFractions) works when limited to odd denominators.

Conjecture 1.

For any fraction 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.

Title	conjecture on fractions with odd denominators
Canonical name	ConjectureOnFractionsWithOddDenominators
Date of creation	2013-03-22 12:48:34
Last modified on	2013-03-22 12:48:34
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Conjecture
Classification	msc 11D68
Classification	msc 11A67
Related topic	SierpinskiErdosEgyptianFractionConjecture