Adam has a secret natural number $x$ which Eve is trying to discover. At each stage Eve may only ask questions of the form "is $x+n$ a prime number?" for some natural number $n$ of her choice.

Prove that Eve may discover $x$ using finitely many questions.
1. **Initial Assumption and Setup:**
   - We start by assuming that \( x \) is a prime number. If \( x \) is not initially prime, we can find a prime number greater than \( x \) by asking if \( x+1, x+2, \ldots \) are prime until we get a "yes" answer. Let this prime be \( x+c \). We then set \( x+c \) as our new \( x \).

2. **Using Dirichlet's Theorem:**
   - Dirichlet's theorem states that there are infinitely many primes of the form \( a + nd \) if \( \gcd(a, d) = 1 \). This will be useful in constructing our questions.

3. **Constructing the Questions:**
   - For each prime \( p \), we need to find prime numbers \( q_1, q_2, \ldots, q_{p-1} \) such that \( q_i \equiv i \pmod{p} \) and \( q_i > 2p \). This is possible due to Dirichlet's theorem.

4. **Formulating the Questions:**
   - We ask if \( x + q_i - p \) is prime for each \( i \) from 1 to \( p-1 \).

5. **Analyzing the Responses:**
   - If \( x = p \), then \( x + q_i - p = q_i \), which is prime by construction. Thus, we will get "yes" answers for all \( i \).
   - If \( x \neq p \), then \( x \equiv -i \pmod{p} \) for some \( i \) in \( 1, \ldots, p-1 \). This means \( p \mid (x + q_i - p) \), and since \( x + q_i - p > p \), it is not prime. Therefore, we will get at least one "no" answer.

6. **Iterating Over Primes:**
   - We repeat the above process for increasing values of \( p \) until we get all "yes" answers for a particular \( p \). When this happens, we can conclude that \( x = p \).

Conclusion:
By iterating over all primes \( p \) and using the responses to our questions, Eve can determine the value of \( x \) in finitely many steps.

The final answer is \( \boxed{ x } \).