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: \begin{equation*}\frac{a}{b} - \frac{c}{d} = \frac{1}{n}. \end{equation*} For example the two proper fractions and unit fractions $\frac{1}{11}$ and $\frac{1}{12}$ are adjacent since: \begin{equation*} \frac{1}{11} - \frac{1}{12} = \frac{1}{132} \; . \end{equation*} $\frac{1}{17}$ 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:

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