Editing Practical number (section)

==Practical numbers and Egyptian fractions==
If <math>n</math> is practical, then any [[rational number]] of the form <math>m/n</math> with <math>m<n</math> may be represented as a sum <math display=inline>\sum d_i/n</math> where each <math>d_i</math> is a distinct divisor of <math>n</math>. Each term in this sum simplifies to a [[unit fraction]], so such a sum provides a representation of <math>m/n</math> as an [[Egyptian fraction]]. For instance,
<math display=block>\frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac12+\frac1{10}+\frac1{20}.</math>

Fibonacci, in his 1202 book ''[[Liber Abaci]]''<ref name="sigler"/> lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above. This method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100.

{{harvtxt|Vose|1985}} showed that every rational number <math>x/y</math> has an Egyptian fraction representation with <math>O(\sqrt{\log y})</math> terms. The proof involves finding a sequence of practical numbers <math>n_i</math> with the property that every number less than <math>n_i</math> may be written as a sum of <math>O(\sqrt{\log n_{i-1}})</math> distinct divisors of <math>n_i</math>. Then, <math>i</math> is chosen so that <math>n_{i-1}<y<n_i</math>, and <math>xn_i</math> is divided by <math>y</math> giving quotient <math>q</math> and remainder <math>r</math>. It follows from these choices that <math>\frac{x}{y}=\frac{q}{n_i}+\frac{r}{yn_i}</math>. Expanding both numerators on the right hand side of this formula into sums of divisors of <math>n_i</math> results in the desired Egyptian fraction representation. {{harvtxt|Tenenbaum|Yokota|1990}} use a similar technique involving a different sequence of practical numbers to show that every rational number <math>x/y</math> has an Egyptian fraction representation in which the largest denominator is <math>O(y\log^2 y/\log\log y)</math>.

According to a September 2015 conjecture by [[Zhi-Wei Sun]],<ref>{{citation |first=Zhi-Wei|last=Sun|url=http://maths.nju.edu.cn/~zwsun/UnitFraction.pdf |title=A Conjecture on Unit Fractions Involving Primes |access-date=2016-11-22 |archive-date=2018-10-19 |archive-url=https://web.archive.org/web/20181019140138/http://maths.nju.edu.cn/~zwsun/UnitFraction.pdf |url-status=dead }}</ref> every positive rational number has an Egyptian fraction representation in which every denominator is a practical number. The conjecture was proved by {{harvs|first=David|last=Eppstein|author-link=David Eppstein|year=2021|txt}}.