Let
![a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^+](/media/m/b/e/e/beeb9dde5a6d881fb25ce72506c1a0c1.png)
be given and let N
![(a_1, a_2, a_3)](/media/m/4/4/f/44f3af500aa6b523cd09f5be21c2b3a7.png)
be the number of solutions
![(x_1, x_2, x_3)](/media/m/6/2/5/62566c2616d4993eaff8e05d7c92a4a3.png)
of the equation
where
![x_1, x_2,](/media/m/3/0/0/3004052e3b02d1095b01ea7866e1157f.png)
and
![x_3](/media/m/4/3/a/43ae0683e5a02ed52f93b676eaa94cb4.png)
are positive integers. Prove that
%V0
Let $a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^+$ be given and let N$(a_1, a_2, a_3)$ be the number of solutions $(x_1, x_2, x_3)$ of the equation
$$\sum^3_{k=1} \frac{a_k}{x_k} = 1.$$
where $x_1, x_2,$ and $x_3$ are positive integers. Prove that $$N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 + ln(2 a_1)).$$