Euclid's proof of the infinitude of primes

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Euclid's proof of the infinitude of primes is a proof by Euclid that the number of prime numbers is infinite.


We proceed by contradiction. Suppose there are only finitely many prime numbers; let's call them $p_1, p_2, p_3, \ldots, p_n$. Let $x=p_1\cdot p_2\cdot p_3 \cdots p_n + 1$. When $x$ is divided by any of our primes $p_1, p_2, p_3, \ldots, p_n$ it leaves a remainder of 1, so none of these primes divide $x$. Since every positive integer has at least one prime factor, $x$ has some prime factor (possibly itself) not in the set $\{ p_1,p_2,p_3,\ldots,p_n\}$. Thus $\{ p_1,p_2,p_3,\ldots, p_n\}$ does not contain all prime numbers. Contradiction! Our original assumption must have been false, so there are in fact infinitely many primes.

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