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