uclid’s proof of the Infinitude of Primes — Proof by contradiction (Reductio ad absurdum): (1.3.3)
Prove that there are infinite number of prime numbers.
Assume that there are finite many primes:
T = {P1, P2, P3, P4, P5, P6… Pn}
Let Q be a number which is equivalent to the product of the finite many prime numbers, plus one.
Q = (P1 x P2 x P3 x P4 x P5… Pn) + 1
Therefore, there can only be two possible types of numbers that Q can be, namely a prime number, or a composite number. If Q is a prime, that would mean that there is a new prime number that is not on the list, and that the list previously forged is incomplete.
However, if Q is not a prime, it is a number divisible by a prime number, as all whole numbers are made up of prime numbers multiplied together. In this case, assume Q is divisible by the prime number p. By looking at:
Q = (P1 x P2 x P3 x P4 x P5… Pn) + 1
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Whether Q is a prime number or a composite number, the original list is incomplete. As long as the list of prime numbers is finite, it is always possible to find one more new prime number. Therefore, if prime numbers are a finite list, it will always be complete. This then contradicts the original assumption that there are finitely many primes and hence, proves Euclid’s theorem, that the set of prime numbers is
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"The Prime Ministers of Canada - Lester B. Pearson Biography." The Prime Ministers of Canada - Lester B. Pearson Biography. N.p., n.d. Web. 23 Jan. 2014. .
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Pearson, Lester B. Mike; the Memoirs of the Right Honourable Lester B. Pearson. Vol. 1. Toronto:
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