To solve this problem, we need to determine which positive integers can be painted based on the given rules. We start by understanding the concept of a *guerrera* pair and then proceed to analyze the painting process.

1. **Definition of a *guerrera* pair**:
A pair of positive integers $(m, n)$ is called *guerrera* if there exist positive integers $a, b, c, d$ such that:
$$m = ab, \quad n = cd, \quad \text{and} \quad a + b = c + d$$
For example, $(8, 9)$ is a *guerrera* pair because $8 = 4 \cdot 2$, $9 = 3 \cdot 3$, and $4 + 2 = 3 + 3$.

2. Initial painted numbers:
We start by painting the numbers $3$ and $5$.

3. Painting process:
If a positive integer $x$ is not painted and there exists a painted positive integer $y$ such that $(x, y)$ is a *guerrera* pair, then we paint $x$.

4. Constructing the triangle:
We construct a triangle where the $i$-th row consists of integers $n$ such that there exist $a, b \in \mathbb{Z}^+$ satisfying $a + b = i + 1$. This means the $i$-th row contains all products $ab$ where $a + b = i + 1$.

5. Painting the rows:
- The third row contains $3$ (since $3 = 1 \cdot 3$ and $1 + 3 = 4$).
- The fourth row contains $4$ (since $4 = 2 \cdot 2$ and $2 + 2 = 4$).
- The fifth row contains $5$ (since $5 = 1 \cdot 5$ and $1 + 5 = 6$).

6. **General case for $i \geq 6$**:
- If $i$ is composite, then $i = ab$ with $a, b > 1$. Since $a + b \leq i$, there is an earlier row containing $i$.
- If $i$ is prime, then $i - 1$ is not prime. The second entry in the row, $2(i-1)$, can be factored as $ab$ with $a, b > 2$ whenever $i \geq 7$. Thus, $i = 2 + (i - 2) \leq a + b$.

7. Inductive argument:
Suppose all rows less than $i$ are painted (excluding the first and second rows). Then some element of row $i$ must be painted, and hence row $i$ will also get painted.

8. Conclusion:
By induction, all positive integers $x \geq 3$ will eventually be painted. The only integers that cannot be painted are $1$ and $2$.

The final answer is $\boxed{\mathbb{Z}^+ \setminus \{1, 2\}}$.