Question: The purpose of this problem is to determine how many prime numbers there are. Suppose there are a total of (n) prime numbers, and we

The purpose of this problem is to determine how many prime numbers there are. Suppose there are a total of $n$ prime numbers, and we list these in order: \(p_{1}=2

a. Define $X=1+p_{1} p_{2} \ldots p_{n}$. That is, $X$ is equal to one plus the product of all the primes. Can we find a prime number $P_{m}$ that divides $X$ ?

b. What can you say about $m$ ?

c. Deduce that the total number of primes cannot be finite.

d. Show that $P_{n+1} \leq 1+p_{1} p_{2} \ldots p_{n}$.

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