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.

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Music Credits:

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Sources and Citations:

1) Gunter Harder and Don Zagier, 'The Conjecture of Birch and Swinnerton-Dyer' (https://people.mpim-bonn.mpg.de/zagie...)

2) Popular Talk by Manjul Bhargava in the Clay Mathematics Institute, 2016 (   • What is the Birch and Swinnerton-Dyer...  )

3) Notes on Bhargava's talk (https://www.claymath.org/sites/defaul...)

4) Wikipedia contributors, "Birch and Swinnerton-Dyer conjecture," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?...

5) Stewart, Ian, 'Visions of Infinity: The Great Mathematical Problems' (Basic Books, 05-Mar-2013)

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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