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 AdjacentFraction 2013-03-22 12:48:23 2013-03-22 12:48:23 XJamRastafire (349) XJamRastafire (349) 17 XJamRastafire (349) Definition msc 11A67 FareySequence UnitFraction ContinuedFraction NumeratorAndDenominatorIncreasedBySameAmount