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# 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:

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

## Mathematics Subject Classification

11A67*no label found*

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