The Birch and Swinnerton-Dyer Conjecture is a millennium prize problem, one of the famed seven placed by the Clay Mathematical Institute in the year 2000. As the only number-theoretic problem in the list apart from the Riemann Hypothesis, the BSD Conjecture has been haunting mathematicians ever since its conception in the mid-20th century, and relates to several different concepts and applications, including Elliptic Curve Cryptography. The BSD Conjecture is essentially an attempt to determine rational solutions of elliptic curves through a variety of different approaches, including but not limited to L-functions, statistics, complex analysis and rank determination. In this video I attempt to explain the BSD Conjecture in layman terms without shying away from the inner beauty of the equations. Slight error at 14:14, the equation should be y^3, not y^2. ------------------------------------------------------------------------- Music Credits: Special thanks to E7VN (https://open.spotify.com/artist/6erp85MigVPLLohx5xPxwT) for creating some great original soundtracks. Inspired by Kevin MacLeod Link: https://incompetech.filmmusic.io/song/3918-inspired/ License: http://creativecommons.org/licenses/by/4.0/ Wind Of The Rainforest Preview by Kevin MacLeod Link: https://incompetech.filmmusic.io/song/5729-wind-of-the-rainforest-preview License: http://creativecommons.org/licenses/by/4.0/ Music: https://www.purple-planet.com/ http://www.bensound.com/royalty-free-music -------------------------------------------------------------------------- Sources and Citations: 1) Gunter Harder and Don Zagier, 'The Conjecture of Birch and Swinnerton-Dyer' (https://people.mpim-bonn.mpg.de/zagier/files/tex/BSDwHarder/fulltext.pdf) 2) Popular Talk by Manjul Bhargava in the Clay Mathematics Institute, 2016 (https://www.youtube.com/watch?v=2gbQWIzb6Dg) 3) Notes on Bhargava's talk (https://www.claymath.org/sites/default/files/bhargava.pdf) 4) Wikipedia contributors, "Birch and Swinnerton-Dyer conjecture," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Birch_and_Swinnerton-Dyer_conjecture&oldid=948783592 5) Stewart, Ian, 'Visions of Infinity: The Great Mathematical Problems' (Basic Books, 05-Mar-2013) -------------------------------------------------------------------------- Chapters 0:00 Introduction to Elliptic Curves 1:40 Intro 1:50 What is Number Theory? 3:27 Types of Diophantine Equations 6:22 Solutions to Elliptic Curves 11:50 Clock Arithmetic 13:13 Analyzing Solutions using Modular Arithmetic 15:45 The Conjecture 17:45 What's been done? 19:03 Conclusion