I am doing an assignment about Egyptian fractions and I am a bit confused about what to do when the given fraction's numerator is greater than denominator. My initial idea was to subtract the fraction by 1, 1/2, 1/3 etc and when the numerator becomes less than denominator I would apply the proper algorithm. And I saw that this way does not work out. Do you have any suggestions?
Thank you.
EDIT: I cannot subtract by 1 because it will not be an Egyptian fraction.
Your trick is pretty close to getting what you want, but you stop too early. As you probably know, the series $$ 1 + \frac12 + \frac13 + \frac14 + \cdots $$ diverges, which means that for any initial value $\frac pq$ you started with, there is an $n$ such that $$ H_n = 1 + \frac12 + \cdots + \frac1n \leq \frac pq < 1 + \frac12 + \cdots + \frac1{n+1} = H_{n+1}. $$ Now, you can look at the rational $\frac pq - H_n$, which must be less than $\frac1{n+1}$. Thus, if we apply any ordinary Egyptian fraction algorithm that works for fractions less than 1, you get a representation of $\frac pq - H_n$, and any term in this Egyptian fraction decomposition will have denominator greater than $n + 1$. Now just add $H_n$ back to this representation, and you have a decomposition of $\frac pq$ of the form that you are looking for.