# theory and/or algorithm to express a proper fraction ...

### theory and/or algorithm to express a proper fraction ...

as the addition of proper fractional terms which denominators are single prime numbers and their powers.

Notice that I am not exactly talking here about aliquot ratios:

https://www.encyclopediaofmath.org/inde ... quot_ratio

Yet, I don't know if those two problems are essentially related (I suspect not).

For instance: 60 = (2^2)(3)(5), and

(29/60) = (1/2) - (3/4) + (1/3) + (2/5)

How do you find the numerators, that is: 1, -3, 1 and 2?

For fractions which denominators have simple prime factorizations you can see what the numerators should be, but I have never found a complete theory about such rewriting of proper fractions which numerator and denominator are coprimes as additions of proper fractions which denominators are the factors belonging to the denominator's prime factorization.
jweinberg

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