Intellectually Curious

Intellectually Curious is a podcast by Mike Breault featuring over 1,400 AI-powered explorations across science, mathematics, philosophy, and personal growth. Each short-form episode is generated, refined, and published with the help of large language models—turning curiosity into an ongoing audio encyclopedia. Designed for anyone who loves learning, it offers quick dives into everything from combinatorics and cryptography to systems thinking and psychology.

Inspiration for this podcast:

"Muad'Dib learned rapidly because his first training was in how to learn. And the first lesson of all was the basic trust that he could learn. It's shocking to find how many people do not believe they can learn, and how many more believe learning to be difficult. Muad'Dib knew that every experience carries its lesson."

― Frank Herbert, Dune

Note: These podcasts were made with NotebookLM. AI can make mistakes. Please double-check any critical information.

All Episodes

Intellectually Curious

BSD Unveiled: Elliptic Curves, L-Functions, and the Million-Dollar Question

October 20, 2025 • Mike Breault

We take a guided tour of the Birch and Swinnerton-Dyer conjecture, the deep link between rational points on elliptic curves and the analytic behavior of L-functions. From Mordell’s theorem on finite bases for rational points, through the enigmatic Tate-Shafarevich group, to how the order of vanishing of the L-function at s = 1 encodes rank, and the practical consequences like Tunnell’s congruent-number test under BSD. We also summarize what is known, what remains open (especially for higher rank curves), and why this bridge between algebra and analysis remains a central frontier in number theory.

Note: This podcast was AI-generated, and sometimes AI can make mistakes. Please double-check any critical information.