# adjacent fraction

Two fractions $\frac{a}{b}$ and $\frac{c}{d}$, $\frac{a}{b}>\frac{c}{d}$ of the positive integers $a,b,c,d$ are adjacent if their difference is some unit fraction^{} $\frac{1}{n}$, $n>0$ that is, if we can write:

$$\frac{a}{b}-\frac{c}{d}=\frac{1}{n}.$$ |

For example the two proper fractions and unit fractions $\frac{1}{11}$ and $\frac{1}{12}$ are adjacent since:

$$\frac{1}{11}-\frac{1}{12}=\frac{1}{132}.$$ |

$\frac{1}{17}$ and $\frac{1}{19}$ are not since:

$$\frac{1}{17}-\frac{1}{19}=\frac{2}{323}.$$ |

It is not necessary of course that fractions are both proper fractions:

$$\frac{20}{19}-\frac{19}{19}=\frac{1}{19}.$$ |

or unit fractions:

$$\frac{3}{4}-\frac{2}{3}=\frac{1}{12}.$$ |

All successive terms of some Farey sequence ${F}_{n}$ of a degree $n$ are always adjacent fractions. In the first Farey sequence ${F}_{1}$ of a degree 1 there are only two adjacent fractions, namely $\frac{1}{1}$ and $\frac{0}{1}$.

Adjacent unit fractions can be parts of many Egyptian fractions^{}:

$$\frac{1}{70}+\frac{1}{71}=\frac{141}{4970}.$$ |

Title | adjacent fraction |
---|---|

Canonical name | AdjacentFraction |

Date of creation | 2013-03-22 12:48:23 |

Last modified on | 2013-03-22 12:48:23 |

Owner | XJamRastafire (349) |

Last modified by | XJamRastafire (349) |

Numerical id | 17 |

Author | XJamRastafire (349) |

Entry type | Definition |

Classification | msc 11A67 |

Related topic | FareySequence |

Related topic | UnitFraction |

Related topic | ContinuedFraction |

Related topic | NumeratorAndDenominatorIncreasedBySameAmount |