# Aegyptian fractions

It is fractions whose numerator is 1 because it is the only ones which were used in
ancient Aegypt. They are also called unit fractions.
Then 2/3 is not an Aegyptian fraction,
but we can represent 2/3 as a

**sum** of Aegyptian fractions, for instance

2/3 = 1/2 + 1/6.
There are many representations of 2/3.

Relation 1/a = 1/(a+1) + 1/(a(a+1)) gives an infinite number of them.

1/2 = 1/3 + 1/6 gives 2/3 = (1/3 + 1/6) + 1/6 and because

1/6 = 1/7 + 1/42 we get

2/3 = 1/3 + 1/6 + 1/7 + 1/42.
Because the harmonic series 1 + 1/2 + 1/3 + 1/4 +...+ 1/n is divergent,
any number, even >1, can be represented as a sum of Aegyptian fractions.
But the number of terms becomes then very high and we restrict to

0<x<1.
We can also search for a "normal representation" with some added constraints.

The simplest seems to choose at every step the largest fraction as possible, that is the lowest denominator.
x = 1/a + r with minimum a and 0≤r<1, gives
a = ⌈1/x⌉,
that is the lowest number ≥ 1/x.
Continue with the remainder r, taking ⌈1/r⌉

Example 4/17 : 17/4 = 5 in excess, remains 4/17 - 1/5 = 3/(5×17).
The next term is (5×17)/3 = 29 in excess.

remains 3/(5×17) - 1/29 = 2/(5×17×29) and next term
(5×17×29)/2 = 1233 in excess, then remains :

2/(5×17×29) - 1/1233 = 1/(5×17×29×1233) which is an Aegyptian fraction, and that's all :

4/17 = 1/5 + 1/29 + 1/1233 + 1/3039345.

Note that it is not always the most "simple" representation :
4/17 = 1/5 + 1/30 + 1/510 with just 3 terms, and highest denominator 510.

## Application

As a puzzle application, note the

camels problem equivallent
to writing 1 as a sum of Aegyptian fractions.

For instance 1 = 1/2 + 1/4 + 1/5 + 1/20 giving 19 camels to be shared in 3 heirs,
having 1/2, 1/4 and 1/5 of the 19 camels.

A program for calculating Aegyptian fractions.

## Engel series

We can write 4/17 = 1/5 + 3/(5×17) = 1/5 × (1+3/17),
decomposition of 3/17 gives

3/17 = 1/6 × (1+1/17) and finally :

4/17 = 1/5 + 1/(5×6) + 1/(5×6×17) that is 1/5 + 1/30 + 1/510,
that time being the simplest one for this example.

An "Engel series" is the representation of a number as :

x = 1/a_{1} × (1 + 1/a_{2} × (1 + 1/a_{3} × (1 + 1/a_{4} (...
= 1/a_{1} + 1/(a_{1}a_{2}) + 1/(a_{1}a_{2}a_{3}) + 1/(a_{1}a_{2}a_{3}a_{4}) + ...
with a_{1}≥2 and a_{n+1}≥a_{n}

It is in that form that we consider the "normal decomposition" in sum of Aegyptian fractions
(normal unit series).

## Sylvester series

There are many other ways of "normalizing" the unit series,
for instance as Sylvester series. .../...