The purpose of this problem is to determine how many prime numbers there are. Suppose there are
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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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a We are assuming that pn is the largest of all primes Because ...View the full answer
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