| 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 +p1p2...pn. That is, X is equal to one plus the product of all the primes. Can we find 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 + p1p2...pn. | |
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