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My apologies if the question has already been discussed somewhere else, I did not found anything related to unitary fractions with the search tool...

It is a nice exercise for high-school students to prove that any positive rational number less than 1 can be written as a sum of unitary fractions with distinct denominators. One possible proof is to consider the lesser integer n such that p/q-1/n is positive; we obtain a fraction whose numerator is np-q which can be at most p-1.

So I was considering the following questions :

  • in some cases, since nq-p can be equal to p-1, it seems possible that one cannot write p/q as a sum of less than p unitary fractions. Is it true that one can find such a fraction for any integer p ?

  • finding a sum of unitary fractions which is equal to a fraction p/q is not difficult, but is it easy to find the sum with a minimum number of fractions ? And the sum which minimize the sum of denominators ?

Thanks by advance for any hint or reference concerning those questions! (Also, I would be very interested in questions related to this topic that could be solved at high-school level)

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Do you mean $np-q$? Because $p/q-1/n = (np-q)/nq$. – Sean Eberhard Mar 14 '12 at 13:33
Thanks Sean, that was a mistake, it is np-q as you noticed it. Thanks Charles for the references which solve the first question; however I haven't found anything concerning the minimal sum of denominators. – Nekochan Mar 14 '12 at 14:20

I doubt that there are nice algorithms for the second question. It could be described as having two parts: the minimum denominator sum (mds) question and the minimal numerator sum (mns) question. For the first, we know one sum for a given rational and then there are an enormous but finite number of sums with a smaller sum of denominators. So the mds question can be answered in finite time for any rational $0 \lt r \lt 1$, but perhaps not in any reasonable manner (Although obviously not all these sums need to be examined.)

I'm not sure that there is an algorithm for the mns question which is sure to provide an answer for the mns question in finite time. It is conjectured that every fraction $\frac{4}{n}$ can be written as the sum of at most three unit fractions. The first counter-example , if there is any, is over $10^{14}$, has a prime denominator, and that denominator belongs to one of 6 congruence classes$ \mod{840}$ but it is an open problem. This means that there is not (known to be) an easy way to find the minimum number of fractions.

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