Suppose you have a rational number x given by:
$$ x = \frac pq $$
Then given an error bound $ \epsilon $ give x as:
$$ x - \sum_{i=1}^n \frac{j_i}{k_i} \le \epsilon $$
where n is minimized and j,k are prime.
I thought about this problem while working on something in Computer Science. Even if a solution is not possible I would appreciate how to think and solve this problem (preferably efficiently).
Edit1: People have pointed out that for arbitrarily large j,k this is always possible. To be more clear rather than minimize n we want to minimize the Kolmogorov complexity of the series : $\sum_{i=1}^n \frac{j_i}{k_i} $