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/a1 × (1 + 1/a2 × (1 + 1/a3 × (1 + 1/a4 (... = 1/a1 + 1/(a1a2) + 1/(a1a2a3) + 1/(a1a2a3a4) + ... with a1≥2 and an+1≥an
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. .../...

 

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