Editing Proof by contradiction (section)

===Infinitude of primes===

Let us take a second look at [[Euclid's theorem]] – Book IX, Proposition 20:<ref name="mathcs_clarku_edu" />

: Prime numbers are more than any assigned multitude of prime numbers.

We may read the statement as saying that for every finite list of primes, there is another prime not on that list,
which is arguably closer to and in the same spirit as Euclid's original formulation. In this case [[Euclid's theorem#Euclid's proof|Euclid's proof]] applies refutation by contradiction at one step, as follows.

Given any finite list of prime numbers <math>p_1, \ldots, p_n</math>, it will be shown that at least one additional prime number not in this list exists. Let <math>P = p_1 \cdot p_2 \cdots p_n</math> be the product of all the listed primes and <math>p</math> a [[prime factor]] of <math>P + 1</math>, possibly <math>P + 1</math> itself. We claim that <math>p</math> is not in the given list of primes. Suppose to the contrary that it were (an application of refutation by contradiction). Then <math>p</math> would divide both <math>P</math> and  <math>P + 1</math>, therefore also their difference, which is <math>1</math>. This gives a contradiction, since no prime number divides 1.