# 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 |