It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions).
I am struggling to understand in a formal way, what algebraic structure such a set of Egyptian fractions of a positive rational is, and of what algebraic properties?
Thanks in advance and references are also welcome.
In fact, the rational number are rational because they are a fractions of two numbers. it may be different with positive integers or the sum of egyptians fractions which are link by a rule.
$1/2 = 1/3 + 1/6$
$1/3 = 1/4 + 1/12$
$1/4 = 1/5 + 1/20$
$1/5 = 1/6 + 1/30$
$1/6 = 1/7 + 1/42$
so we can say $1/U(1)=1/U(0)-1/U(0)U(1)$
$U(1)/U(0)-U(1)/U(0)U(1)=1 $ if $ U(1)=U(0)+1$
it is in french, but you can study the egyptian fraction (and the code) in clouds: http://jeux-et-mathematiques.davalan.org/arit/egy/index.html#table2
There are the same question here. A question about rational.