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Let's fill in the blanks for the proof in Problem 9 based on the information provided and the typical structure of this proof.

### Given Information:
1. **Theorem 1** states that if \( b \) is a natural number, then \( \gcd(b, b+1) = 1 \), meaning \( b \) and \( b+1 \) are coprime.
2. **Theorem 2** provides a criterion for an infinite set: if for each finite subset \( S \) of a set \( B \), there exists an element \( x \in B \) such that \( x \notin S \), then \( B \) is infinite.
3. **Theorem 3** is the statement we need to prove, that there are infinitely many prime numbers.

### Proof Outline with Filled Blanks:
Let's proceed step-by-step and fill each blank in the proof.

1. **Let \( \mathbb{P} \) be the set of** prime numbers.
   - This is the set of all primes, which we are trying to show is infinite.

2. **Let \( Q \) be a** finite subset **of the set \( \mathbb{P} \). Denote the elements of \( Q \) by \( p_1, p_2, \dots, p_n \) and let \( q = p_1 \cdot p_2 \cdots p_n \).**
   - Here, we consider a finite subset \( Q \) of prime numbers and define \( q \) as the product of all primes in \( Q \).

3. **By Theorem 1, \( q \) and \( q + 1 \) are** coprime.
   - According to Theorem 1, \( q \) (product of primes in \( Q \)) and \( q+1 \) are coprime.

4. **So there is at least one prime number that** divides **\( q + 1 \) but** not **\( q \).**
   - There must be a prime factor of \( q+1 \) that is not in the original set \( Q \), because \( q+1 \) is not divisible by any of the primes in \( Q \).

5. **Let's call this prime number \( t \). Because \( t \) does not divide \( q \), we have that \( t \) is** not an element **of \( Q \).**
   - This new prime \( t \) divides \( q+1 \), proving that there is always a prime not in \( Q \).

6. **So we have shown that for any finite set of prime numbers \( Q \), we can find another prime number that is not in the set \( Q \). Thus, by Theorem 2, we have that \( \mathbb{P} \) is** infinite.
   - By Theorem 2, since we can always find a new prime outside any finite subset, the set of primes \( \mathbb{P} \) must be infinite.

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### Final Answer for Each Blank:
1. \( \mathbb{P} \) = prime numbers.
2. \( Q \) = finite subset.
3. coprime.
4. divides.
5. not.
6. infinite.

Would you like additional details on any part of this proof or have any questions?

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Here are some related questions to expand your understanding:
1. Why is \( q+1 \) guaranteed to have a prime factor not in \( Q \)?
2. How does Theorem 2 guarantee the infiniteness of \( \mathbb{P} \)?
3. Can you explain why \( q \) and \( q+1 \) are coprime for any finite set \( Q \)?
4. What would happen if there were only finitely many primes?
5. How does this proof relate to Euclid’s original proof of the infiniteness of primes?

**Tip:** When proving a set is infinite, demonstrating that new elements can always be found outside any finite subset is often a strong approach.

Prove that there are infinitely many prime numbers using Theorems 1 and 2.

Learn how to prove that there are infinitely many prime numbers using coprime properties and set theory principles. This problem uses Theorem 1 (coprime property of consecutive numbers) and Theorem 2 (infinite set criterion) to demonstrate that prime numbers are infinite. Suitable for college-level number theory.

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