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 $0\\leq\\frac{a}{2k+1}<1$ 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.

单词	ConjectureOnFractionsWithOddDenominators
释义	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 $0\\leq\\frac{a}{2k+1}<1$ 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.
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