| 8.1 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: p1 = 2 < p2 = 3 < p3 = 5 <...< pn. a. Define X = 1 + p1 p2...pn. That is, X is equal to one plus the product of all the primes. Can we find a prime number Pm that divides X? b. What can you say about m? c. Deduce that the total number of primes cannot be finite. d. Show that Pn+1 . 1 + p1 p2..pn. | |
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