Unsolved problems in number theory | Number theory | Conjectures

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers a, b and c (hence the name) that are relatively prime and satisfy a + b = c. The conjecture essentially states that the product of the distinct prime factors of abc is usually not much smaller than c. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis". The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture. Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still regarded as unproven. (Wikipedia).

ABC Intro - part 1 - What is the ABC conjecture?

This videos gives the basic statement of the ABC conjecture. It also gives some of the consequences.

From playlist ABC Conjecture Introduction

Undergraduate math talk: The abc conjecture

This mathematics talk is an introduction to the abc conjecture. The talk explains what the abc conjecture is, gives a proof of an analog for polynomials, and finishes with some comments on the proof for integers.

From playlist Math talks

The ABC Conjecture, Brian Conrad (Stanford) [2013]

slides for this talk: https://drive.google.com/file/d/1J04zXCQYgn9MdgDUo63rH719cruiQJVo/view?usp=sharing The ABC Conjecture Brian Conrad [Stanford University] Stony Brook Mathematics Colloquium Video September 12, 2013 http://www.math.stonybrook.edu/Videos/Colloquium/video_slides.php?

From playlist Number Theory

An introduction to the abc conjecture - Héctor Pastén Vásquez

Members’ Seminar Topic: An introduction to the abc conjecture Speaker: Héctor Pastén Vásquez Date: Monday, March 21 In this talk I will discuss some classical and new applications of the abc conjecture, its relation to conjectures about elliptic curves, and some (admittedly weak) uncon

From playlist Mathematics

Cameron L. Stewart: A refinement of the abc conjecture

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Number Theory

IUT overview: What papers are involved? Where does it start?

In this video I give an overview of what papers are involved in Mochizuki's work on ABC. Hopefully this is useful to get a scope of things.

From playlist ABC Conjecture Introduction

ABC Intro - part 2 - Frey Curve, Szpiro, and ABC

In this video we introduce Szpiro's Conjecture and show how it implies the ABC conjecture.

From playlist ABC Conjecture Introduction

What is the Riemann Hypothesis?

This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation

From playlist Mathematics

In this video, we explore the "pattern" to prime numbers. I go over the Euler product formula, the prime number theorem and the connection between the Riemann zeta function and primes. Here's a video on a similar topic by Numberphile if you're interested: https://youtu.be/uvMGZb0Suyc The

From playlist Other Math Videos

Shimura curves and new abc bounds -Hector Pasten

Joint IAS/Princeton University Number Theory Seminar Topic: Shimura curves and new abc bounds Speaker: Hector Pasten Affiliation: Harvard University Date: November 28, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Héctor H. Pastén Vásquez: Shimura curves and bounds for the abc conjecture

Abstract: I will explain some new connections between the abc conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the abc conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as

From playlist Algebraic and Complex Geometry

Gary Walsh: On binary quartic Thue equations and related topics

CIRM VIRTUAL CONFERENCE Recorded during the meeting " Diophantine Problems, Determinism and Randomness" the November 25, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide

From playlist Virtual Conference

A (compelling?) reason for the Riemann Hypothesis to be true #SOME2

A visual walkthrough of the Riemann Zeta function and a claim of a good reason for the truth of the Riemann Hypothesis. This is not a formal proof but I believe the line of argument could lead to a formal proof.

From playlist Summer of Math Exposition 2 videos