Give a proof for the following statement. There are infinitely many primes. Solu

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proof: Suppose, to the contrary, that there are a finite number of prime numbers. Let p1, p2, p3,…, pk be the entire collection of prime numbers. Define n = (p1p2...pk) + 1, that is, n is equal to the product of every prime number plus one. The n is a natural number, and by the Fundamental Theorem of Algebra, n can be expressed uniquely as a product of primes. That implies that n has a prime divisor q. Then q > 1. And, since q is prime, q is in the collection of all primes, and therefore, q divides the product (p1p2...pk). Since q divides both n and the product of all the primes, it divides their difference, i.e., n - (p1p2...pk) = 1. But the only divisor of 1 is 1. So q = 1. Then we have q = 1 and q > 1, which is a contradiction. We conclude that our supposition that the primes were finite in number is false, and therefore, there must exist infinitely many prime numbers.

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