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Currently Playing: What was Fermat’s “Marvelous" Proof? | Infinite Series

Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi If Fermat had a little more room in his margin, what proof would he have written there? Tweet at us! @pbsinfinite Facebook: facebook.com/pbsinfinite series Email us! pbsinfiniteseries [at] gmail [dot] com Resources: Contemporary Abstract Algebra by Joseph Gallian https://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702 Standard Definitions in Ring Theory by Keith Conrad http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/ringdefs.pdf Rings and First Examples (online course by Prof. Matthew Salomone) https://www.youtube.com/watch?v=h4UCMd8dyiM Fermat's Enigma by Simon Singh https://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622 Who was first to differentiate between prime and irreducible elements? (StackExchange) https://hsm.stackexchange.com/questions/3754/who-was-first-to-differentiate-between-prime-and-irreducible-elements Previous Episodes: What Does It Mean to be a Number? https://www.youtube.com/watch?v=3gBoP8jZ1Is What are Numbers Made of? https://www.youtube.com/watch?v=S4zfmcTC5bM Gabe's references from the comments: Blog post about the Peano axioms and construction of natural numbers by Robert Low: http://robjlow.blogspot.co.uk/2018/01/whats-number-1-naturally.html Recommended by a viewer for connections to formulation of numbers in computer science: https://softwarefoundations.cis.upenn.edu/ In 1637, Pierre de Fermat claimed to have the proof to his famous conjecture, but, as the story goes, it was too large to write in the margin of his book. Yet even after Andrew Wiles’s proof more than 300 years later, we’re still left wondering: what proof did Fermat have in mind? The mystery surrounding Fermat’s last theorem may have to do with the way we understand prime numbers. You all know what prime numbers are. An integer greater than 1 is called prime if it has exactly two factors: 1 and itself. In other words, p is prime if whenever you write p as a product of two integers, then one of those integers turns out to be 1. In fact, this definition works for negative integers, too. We simply incorporate -1. But the prime numbers satisfy another definition that maybe you haven’t thought about: An integer p is prime if, whenever p divides a product of two integers, then p divides at least one of those two integers. Written and Hosted by Tai-Danae Bradley Produced by Rusty Ward Graphics by Ray Lux Assistant Editing and Sound Design by Mike Petrow and Linda Huang Made by Kornhaber Brown (www.kornhaberbrown.com) Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon. Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level! And thanks to Mauricio Pacheco who are supporting us at the Lemma level!