Does the Diophantine equation $a/b + c/d = e/f$ have a solution for coprime denominators?

Question

Does the Diophantine equation $\frac a b + \frac c d = \frac e f$ have a solution where:

• The denominators are pairwise coprime and $> 1$
• The numerators are not exact multiples of the denominators, so each fraction is not an integer.

I suppose it can be formulated as $adf + cbf = ebd$ with the same conditions.

Answer 1

First, I ran a brute force search, but I realized that the solution is pretty straight-forward. Suppose there exists some $6$-tuple that satisfies your constraints. Then, as you state, $adf + cbf = ebd$. Therefore, $f \mid ebd$. But, $(d, f) = 1$ and $(b, f) = 1$, so $f \mid e$, but this violates your constraint, leading to a contradiction.

Hence, there are no solutions.