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

Question

(This is a followup question to $\frac a b + \frac c d = \frac e f$ which has a straightforward answer.)

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

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

It can be formulated as $ad + cb = ebd$ with the same conditions.

Answer 1

No. Reduce both fractions until $(a,b)=(c,d)=1$. This keeps the coprimality of $b$ and $d$. You cannot end up with $d=1$ or $b=1$ otherwise the second condition would be not fulfilled. Then $d$ must divide $cb$, which is impossible because it is $>1$ and coprime with both.

Answer 2

No; write $a/b$ in simplest terms. We have $$ \frac ab = \frac{de - c}d, $$ but then $b \mid d$.