\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For \(n \in \mathbb{N}^*\), let \(p_n\) be the \(n\)-th prime number (\(p_1 = 2, p_2 = 3, \dots\)).
The goal of this exercise is to prove that for every \(n \geq 2\):$$p_{n+1} < p_1 \times p_2 \times \dots \times p_n$$Let \(n \geq 2\) and consider the integer \(M = (p_1 \times p_2 \times \dots \times p_n) - 1\).
  1. Justify that \(M\) admits at least one prime divisor \(q\).
  2. Show that for any \(k \in \{1, 2, \dots, n\}\), the prime \(p_k\) does not divide \(M\).
  3. Deduce that \(q > p_n\).
  4. Using the definition of \(p_{n+1}\), show that \(p_{n+1} \leq q\).
  5. Conclude the proof of the inequality.

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