We are tasked with proving the statement:
For each natural number $k$, we call $P_k$ the $k$-th prime. Let $n$ be a natural number with $n \geq 3$. We are to prove that:

$P_n \leq P_1 P_2 \cdots P_{n-1} + 1$

Step-by-Step Proof:

We can use a variation of Euclid's proof of the infinitude of primes. Euclid's proof shows that if we have a finite list of primes, we can construct a number that is either prime or has a prime factor not in the list.

Construct the Number:

Consider the number:

$N = P_1 P_2 \cdots P_{n-1} + 1$

This number $N$ is the product of all the primes up to $P_{n-1}$ plus 1. Now, we analyze the properties of $N$.

Divisibility Check:

We now check whether $N$ is divisible by any of the primes in the list $P_1, P_2, \dots, P_{n-1}$.

• Suppose $P_n$ is one of the primes in the list $P_1, P_2, \dots, P_{n-1}$.
• For each $k \in \{1, 2, \dots, n-1\}$, we have:

$N = P_1 P_2 \cdots P_{n-1} + 1$

• If we divide $N$ by any of $P_1, P_2, \dots, P_{n-1}$, the remainder is always 1:

$P_1 P_2 \cdots P_{n-1} \equiv -1 \pmod{P_k}$

• Therefore, $N$ is not divisible by any of the primes $P_1, P_2, \dots, P_{n-1}$, since dividing by any of these primes gives a remainder of 1.

Conclusion:

Since $N$ is not divisible by any prime in the list $P_1, P_2, \dots, P_{n-1}$, the number $N$ must either be a prime itself or divisible by some prime not in the list. This implies that the prime $P_n$, the next prime after $P_1, P_2, \dots, P_{n-1}$, must satisfy:

$P_n \leq P_1 P_2 \cdots P_{n-1} + 1$

Thus, we have proven the statement.

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Do you need further clarification on any step of the proof or any additional details?
Here are 5 related questions to deepen the understanding of this topic:

1. How does this result connect to Euclid's original proof of the infinitude of primes?
2. What would happen if $n = 2$? Would the result still hold?
3. Can this proof be generalized for any number of primes in the list?
4. How do we define prime numbers in number theory, and why is this result important for understanding their distribution?
5. What is the significance of the number $P_1 P_2 \cdots P_{n-1} + 1$ in the proof, and why does it always lead to finding new primes?

Tip:
This type of proof method—using contradiction and constructing a number that forces the existence of a new prime—appears frequently in number theory, particularly when proving the infinitude of primes or similar results.