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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?

Phi pi hohum
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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 $\frac{1}{p(p+1)}$.

André Nicolas
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  • Thank you for your response. I'm afraid I'm not very good at explaining my aim. I meant something along the lines of 165/224 expressed as 1/2+1/7+1/14+1/112+1/224, only with a prime number as the denominator – Phi pi hohum Aug 29 '14 at 19:57
  • If you can give more detail about what it is exactly that you want, I can (maybe!) help solve the real problem you are after. What is now in the answer is a brief description of an algorithm for expressing a fraction as a sum of "Egyptian" fractions. (There does not seem to be a Hellenic interest in the problem.) – André Nicolas Aug 29 '14 at 20:02
  • The algorithm as described works of $\frac{k}{n}$ whether or not $n$ is prime. Almost always we cannot have the denominators on the right-hand side be primes. – André Nicolas Aug 29 '14 at 20:03
  • i see. that's very interesting. I haven't looked at all at Egyptian fractions. I was dabbling in an old university book about Archimedes' books, and part of the introduction is talking about the old systems of multiplication and fractions. If you google "Books of Archimedes pdf" the top entry should be something from archive.org which gives an online and downloadable version of a scholarly reproduction of his works. The section I'm working with is in the introduction on page 1xx. Much of it is beyond me, but the fraction thing is very fun, but fractions with prime denominators are a problem. – Phi pi hohum Aug 29 '14 at 20:07