Problem 1115

To solve this problem, let's denote the four different natural numbers as $a$, $b$, $c$, and $d$. The condition given is that for any pair of these numbers, say $(a, b)$, the expression:

$$ab + cd$$

must be a prime number. Similarly, for the other pairs, the following expressions must also be prime numbers:

$$ac + bd$$
$$ad + bc$$
$$bc + ad$$

Now, let's try choosing small, distinct natural numbers and checking these conditions. The reference answer provides the numbers 1, 2, 3, and 5. Let's verify:

### Calculations:

1. **Pair $(1, 2)$:**

$$1 \cdot 2 + 3 \cdot 5 = 2 + 15 = 17 \quad (\text{prime})$$

2. **Pair $(1, 3)$:**

$$1 \cdot 3 + 2 \cdot 5 = 3 + 10 = 13 \quad (\text{prime})$$

3. **Pair $(1, 5)$:**

$$1 \cdot 5 + 2 \cdot 3 = 5 + 6 = 11 \quad (\text{prime})$$

4. **Pair $(2, 3)$:**

$$2 \cdot 3 + 1 \cdot 5 = 6 + 5 = 11 \quad (\text{prime})$$

5. **Pair $(2, 5)$:**

$$2 \cdot 5 + 1 \cdot 3 = 10 + 3 = 13 \quad (\text{prime})$$

6. **Pair $(3, 5)$:**

$$3 \cdot 5 + 1 \cdot 2 = 15 + 2 = 17 \quad (\text{prime})$$

Each expression results in a prime number, thus validating our solution. Therefore, a set of four different natural numbers satisfying the given condition is:

$$\boxed{1, 2, 3, 5}$$