Euclid's Proof That Primes Never End

Video source code (manim): https://quantiayt.gumroad.com/ Hand me any finite list of primes. Any list. I will multiply every number on it, add one, and out the other end falls a prime that was not on your list. Every time. No exceptions. A 2300-year-old machine that just keeps printing money. And here is the part that should terrify you. Nobody, in twenty-three centuries of trying, has done it better. Not Euler. Not Gauss. Not Erdos. They all tried. They all left their own proofs behind. And Euclid's is still the one the books put first. When G.H. Hardy sat down in 1940 to explain what beauty in mathematics actually means, he picked exactly two examples. This was one of them. In this video we walk through five proofs of the infinitude of the primes, spanning five centuries: Euclid's original construction from Book Nine (300 BCE), Euler's product formula that invented analytic number theory (1737), Goldbach's one-letter Fermat-number trick (1730), Furstenberg's topological paragraph (1955), and Erdos's squarefree counting argument (1938). Five completely different techniques. One theorem. Same six words every time: this is how it is done. Along the way we unpack the trap almost everyone walks into (no, N is not always prime, and 30031 = 59 x 509 proves it), the identity Euler pulled out of nowhere that bridged integers and primes forever, and the move Euclid makes in Book Nine that every diagonal argument in mathematics ever since has descended from. Cantor, Godel, Euclid. Same move. Three different universes. 00:00 - Hook 00:50 - The Dare in Book Nine 01:45 - The Construction: Multiply, Add One 03:00 - The Trap (Why N Is Not Always Prime) 04:00 - Why Hardy Called It Perfect 05:15 - Euler 1737: The Harmonic Bomb 06:40 - Goldbach 1730: A Letter With Fermat Numbers 07:40 - Furstenberg 1955: Topology Crashes In 09:05 - Erdos 1938: Just Count 10:15 - The Payoff: Cantor, Godel, Euclid #mathematics #numbertheory #euclid #derivia #primes #primenumbers #mathematicalproof #analyticnumbertheory #topology #erdos