# Can every rational number in $(0,1)$ be expressed as a sum or a difference of special fractions?

In a paper, I found the claim that every real number can be expressed as a sum of two so-called $F(4)$-numbers.

These are numbers , for which the entries of the simple continued fraction (except the first number) are at most $4$.

So, the number $[-6,2,3,1]$ is a $F(4)$-number whereas the number $[-1,5,3,2]$ is not because the "$5$" is not in the first position.

I wonder whether every rational number $r$ with $0<r<1$ can be expressed as a sum or difference of fractions $a$ and $b$, such that $a$ and $b$ are $F(4)$-numbers with $0<a,b<1$ (So, the continued fractions start with $0$).

For example , we have $$\frac{29}{430}=\frac{3}{10}-\frac{10}{43}$$ and the continued fractions are $$\frac{3}{10}=[0,3,3]$$ and $$\frac{10}{43}=[0,4,3,3]$$

Hence there is a representation of the required form of $\frac{29}{430}$

Does such a representation (none of the finite continued fractions is allowed to end with the number $1$!) exist for every rational number $r$ with $0<r<1$ ?

If yes, how can I find one (not all) representation efficiently ?

For example, can the number $0.2148324851527661$ be represented as required, and if yes, how ?

• Could you please let us know which paper you are referring to? Thanks! – Matthew Conroy Dec 4 '16 at 19:41
• educ.jmu.edu/~lucassk/Papers/ctac.pdf – Peter Dec 4 '16 at 19:47
• The paper mentions an algorithm, but I cannot find something concrete. – Peter Dec 4 '16 at 19:50