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Full Idea
Assume a largest prime, then multiply the primes together and add one. The new number isn't prime, because we assumed a largest prime; but it can't be divided by a prime, because the remainder is one. So only a larger prime could divide it. Contradiction.
Gist of Idea
An assumption that there is a largest prime leads to a contradiction
Source
report of Euclid (Elements of Geometry [c.290 BCE]) by James Robert Brown - Philosophy of Mathematics Ch.1
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.1
A Reaction
Not only a very elegant mathematical argument, but a model for how much modern logic proceeds, by assuming that the proposition is false, and then deducing a contradiction from it.
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