Problem 1717

1. **Expressing $a$, $b$, and $c$ in terms of relatively prime integers:**
Let
$$a = \frac{k}{\ell}, \quad b = \frac{m}{n}, \quad c = \frac{\ell n}{km}$$
where $\gcd(k, \ell) = \gcd(m, n) = 1$. Given $abc = 1$, we have:
$$\frac{k}{\ell} \cdot \frac{m}{n} \cdot \frac{\ell n}{km} = 1$$
which is true by construction.

2. Formulating the given condition:
We are given that there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Substituting the expressions for $a$, $b$, and $c$, we get:
$$\left(\frac{k}{\ell}\right)^x + \left(\frac{m}{n}\right)^y + \left(\frac{\ell n}{km}\right)^z \in \mathbb{Z}$$
Simplifying, we have:
$$\frac{k^x}{\ell^x} + \frac{m^y}{n^y} + \frac{(\ell n)^z}{(km)^z} \in \mathbb{Z}$$

3. Clearing the denominators:
Multiply through by $\ell^x n^y (km)^z$ to clear the denominators:
$$\frac{k^x \ell^x n^y (km)^z}{\ell^x} + \frac{m^y \ell^x n^y (km)^z}{n^y} + \frac{(\ell n)^z \ell^x n^y (km)^z}{(km)^z} \in \mathbb{Z}$$
Simplifying, we get:
$$k^x m^z n^y + k^z \ell^x m^{y+z} + \ell^{x+z} n^{y+z} \in \mathbb{Z}$$
Dividing by $k^z \ell^x m^z n^y$, we have:
$$\frac{k^{x+z} m^z n^y + k^z \ell^x m^{y+z} + \ell^{x+z} n^{y+z}}{k^z \ell^x m^z n^y} \in \mathbb{Z}$$

4. Analyzing divisibility conditions:
For the above expression to be an integer, the numerator must be divisible by the denominator. This implies:
$$k^z \mid \ell^{x+z} n^{y+z}$$
Since $\gcd(k, \ell) = 1$, it follows that:
$$k^z \mid n^{y+z}$$
Similarly, we obtain the following divisibility conditions:
$$m^z \mid \ell^{x+z} \quad (2)$$
$$\ell^x \mid m^z n^y \quad (3)$$
$$n^y \mid k^z \ell^x \quad (4)$$

5. Handling common prime factors:
Consider a prime $p$ that divides both $\ell$ and $n$. Suppose:
$$p^\alpha \mid \mid \ell, \quad p^\beta \mid \mid n$$
From (3), we have:
$$\alpha x \leq \beta y$$
From (4), we have:
$$\beta y \leq \alpha x$$
Thus, $\alpha x = \beta y$. This implies there exist positive integers $s$ and $t$ such that:
$$\ell = \ell' \cdot s, \quad n = n' \cdot t, \quad t^y = s^x, \quad \gcd(\ell', n') = 1$$

6. Prime factorization and divisibility:
Using the modified versions of (2) and (3), we get:
$$(\ell')^x \mid m^z \quad \text{and} \quad m^z \mid (\ell')^{x+z}$$
This implies $\ell'$ and $m$ have the same set of prime divisors. Let:
$$\ell' = \prod_{j=1}^N p_j^{\alpha_j}, \quad m = \prod_{j=1}^N p_j^{\beta_j}$$
We analyze the $\nu_p$ values for each term in the numerator and denominator.

7. Conclusion:
By analyzing the $\nu_p$ values and ensuring the conditions hold, we conclude that $a$, $b$, and $c$ must be powerful. Specifically, $m$ and $\ell'$ must be nontrivial perfect powers, and similar arguments apply to $k$ and $n'$.