The Infinitude of Primes: Euclid, Euler, Erdös
The foundation of number theory lies among the primes. It thus seems fitting to examine three different proofs, from across history, of the infinitude of the prime numbers.
We first look at Euclid's argument from 300 BCE, which appears as Proposition 20 of Book IX of the Elements. Although "Euclid's proof of the infinitude of primes" is a standard in every number theory textbook, some people might be surprised to see his argument in its original form.
Next, we consider Euler's analytic proof from 1737. Like so much of his work, this features a blizzard of formulas, manipulated with a maximum of agility and a minimum of rigor. But the outcome is spectacular.
Finally, we examine Erdös's combinatorial proof from the 20th century. This is an elementary argument, but it reminds us once again that "elementary" does not mean "trivial."
Taken together, these proofs suggest that, to establish the infinitude of primes, it helps to have a two-syllable last name starting with "E." More to the point, they show mathematics as a subject whose creative variety knows no bounds.
NOTE: The talk is accessible to anyone who has had calculus.