One of the most important facts about prime numbers is that there are infinitely many of them. The proof for this uses a common method for finding mathematical proofs which is contradiction. It firsts assumes that there are a finite number of primes, and then shows why this can not be true.
Proof:
Suppose that p1 = 2 < p2 = 3 < p3 = 5 < ..... < pn are all the prime numbers.
Let P = p1p2p3....pn + 1, and let p be a prime dividing P. p can not be any of the p1,p2,p3,...pn otherwise p would divide the difference P - p1p2p3....pn = 1. No prime number can divide 1, as 1 has only one factor, being itself. Thus, the original assumption must be false. Thus, there are infinite prime numbers.
This will be the final post in the primes series for the time being, but I may return in the future.
Proof:
Suppose that p1 = 2 < p2 = 3 < p3 = 5 < ..... < pn are all the prime numbers.
Let P = p1p2p3....pn + 1, and let p be a prime dividing P. p can not be any of the p1,p2,p3,...pn otherwise p would divide the difference P - p1p2p3....pn = 1. No prime number can divide 1, as 1 has only one factor, being itself. Thus, the original assumption must be false. Thus, there are infinite prime numbers.
This will be the final post in the primes series for the time being, but I may return in the future.
Sorry the text size is off, had some technical difficulties concerning the background colour of the text.
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