So I'm working on practicing reducing fractions into additions of unities (like ancient greek math). It's actually very enjoyable, except when I end up running into a fraction with a prime number as the denominator, as I rely on the multiples of the denominator to extract unities from the fraction and chip away at the numerator until I'm left only with a string of unities, which when added together total the original fraction.
I've tried instilling multiples into the fraction by multiplying by variations of 1 (2/2, 3/3, 4/4, etc...) but this is only met with moderate success.
does anyone have any thoughts on reducing fractions with prime number denominators into strings of unities?
A start: We use the identity $\frac{1}{n}=\frac{1}{n+1}+\frac{1}{(n)(n+1)}$, if necessary repeatedly.
For example, $\frac{2}{p}=\frac{1}{p}+\frac{1}{p+1}+\frac{1}{p(p+1)}$.
If we had $\frac{3}{p}$, as a first step we would get $\frac{3}{p}=\frac{1}{p}+\frac{2}{p+1}+\frac{2}{p(p+1)}$, and we would apply the identity again to one of the $\frac{1}{p+1}$ and to one of the $\frac{1}{p(p+1)}$.