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→The Prime Numbers
==The Prime Numbers==
==The Prime Numbers==
The smallest prime numbers are 2, 3, 5, 7, 11, 13...
The first few prime numbers are 2, 3, 5, 7, 11, 13...
It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.
It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.