Euclid proved that given any finite list of primes, there exists a prime not in the list...

Euclid proved that given any finite list of primes, there exists a prime not in the list. Read the following argument and answer the questions that follow.

Let 2, 3, 5, 7, ... , p be a list of all the primes less than or equal to a certain prime p. We will show that there exists a prime not on the list. Consider the product

Notice that every prime in our list divides that product. However, if we add 1 to the product, that is, form the number N = ( ) + 1, then none of the primes in the list will divide N. Notice that whether N is prime or composite, some prime q must divide N. Because no prime in our list divides N, q is not one of the primes in our list. Consequently q > p. We have shown that there exists a prime greater than p.

a. Explain why no prime in the list will divide N.

b. Explain why some prime must divide N.

c. Someone discovered a prime that has 65,050 digits. How does the preceding argument assure us that there exists an even larger prime?

d. Does the argument show that there are infinitely many primes? Why or why not?

e. Let . Without multiplying, explain why some prime greater than 19 will divide M.