conjecture on fractions with odd denominators


Egyptian fractionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 27 gives 14+128, but we may also write 27 as 17+19+135+1315.

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 0a2k+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.

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