Unsolved problems in number theory | Arithmetic dynamics | Integer sequences | Conjectures
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the 3n + 1 problem, the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered US$500 for its solution. Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". (Wikipedia).
The Collatz Conjecture and Fractals
Visualizing the dynamics of the Collatz Conjecture though fractal self-similarity. Support this channel: https://www.patreon.com/inigoquilez Tutorials on maths and computer graphics: https://iquilezles.org Code for this video: https://www.shadertoy.com/view/llcGDS Donate: http://paypal.m
From playlist Maths Explainers
The Collatz Conjecture: Easy Enough for a 3rd Grader, Hard Enough for Terry Tao
The Collatz Conjecture has been called the simplest math problem no one can prove. It has captivated mathematicians for two generations. In today's Math Minute (the fiftieth Math Minute!), I want to look at just what is so surprising about the Collatz Conjecture, and talk about a few ways
From playlist Math Minutes
Almost all Collatz Orbits Attain Almost Bounded Values - Terence Tao
Members' Colloquium Topic: Almost all Collatz Orbits Attain Almost Bounded Values Speaker: Terence Tao Affiliation: University of California, Los Angeles; Member, School of Mathematics Date: March 13, 2023 Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1
From playlist Mathematics
The Collatz Conjecture - summary of a proof
http://lesliegreen.byethost3.com/articles/neg_collatz.pdf http://lesliegreen.byethost3.com/articles/evidence.zip (now tested to *10 million digits*) The Collatz Conjecture is infamous for being easy to state but impossible to prove by standard methods. Clearly a non-standard approach is re
From playlist Covers
The Collatz conjecture (3n+1 problem) | Famous Math Problems 2 | NJ Wildberger
The Collatz conjecture is tantalizing; simple to state, spectacular in its claim, and notorious for defeating all who attack it. First enunciated by Lothar Collatz in 1937, it has also sometimes been called the Syracuse problem, Kakutani's problem, Ulam's problem, the Hailstone conjecture.
From playlist Famous Math Problems
Collatz Conjecture: MegaFavNumber 63,728,127
My #MegaFavNumbers number is 63,728,127. Link to code: https://github.com/lizard-heart/collatz-ratios Numbers I found: 1, 3, 7, 9, 27, 230631, 626331, 837799, 1723519, 3732423, 6649279, 8400511, 63728127 UPDATE: a number with a lower ratio than 63,728,127 (0.0267... vs 0.0273....) has be
From playlist MegaFavNumbers
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
Dimitri Zvonkine - On two ELSV formulas
The ELSV formula (discovered by Ekedahl, Lando, Shapiro and Vainshtein) is an equality between two numbers. The first one is a Hurwitz number that can be defined as the number of factorizations of a given permutation into transpositions. The second is the integral of a characteristic class
From playlist 4th Itzykson Colloquium - Moduli Spaces and Quantum Curves
Coding in the Cabana 2: Collatz Conjecture
It's the second episode of Coding in the Cabana! Here I attempt to visualize the Collatz Conjecture in Processing. 💻https://thecodingtrain.com/CodingInTheCabana/002-collatz-conjecture.html 🔗 Collatz Conjecture on Wikipedia: https://en.wikipedia.org/wiki/Collatz_conjecture 🔗 PDF Export -
From playlist Coding in the Cabana
Elliptic genera of Pfaffian-Grassmannian double mirrors - Lev Borisov
Lev Borisov Rutgers University November 5, 2014 More videos on http://video.ias.edu
From playlist Mathematics
Collatz Conjecture in Color - Numberphile
The Great Courses Plus (free trial): http://ow.ly/RqOr309wT7v This video features Alex Bellos. More info and links in full description. Extra footage with Alex and coloring: https://youtu.be/w8nc8wbgXPU Or real-time video of the coloring: https://youtu.be/wH141HLD57o Our previous Collatz
From playlist Alex Bellos on Numberphile
*** This is CS50, Harvard University's introduction to the intellectual enterprises of computer science and the art of programming. *** HOW TO SUBSCRIBE http://www.youtube.com/subscription_center?add_user=cs50tv HOW TO TAKE CS50 edX: https://cs50.edx.org/ Harvard Extension School: ht
From playlist CS50 Shorts
Collatz Conjecture (extra footage) - Numberphile
Main video on Collatz Conjecture: https://youtu.be/5mFpVDpKX70 Riemann Hypothesis: https://youtu.be/d6c6uIyieoo Key to the Riemann Hypothesis: https://youtu.be/VTveQ1ndH1c Eisenbud 17-gon: https://youtu.be/87uo2TPrsl8 Fermat's Last Theorem: https://youtu.be/qiNcEguuFSA Bridges to Fermat (K
From playlist David Eisenbud on Numberphile
UNCRACKABLE? The Collatz Conjecture - Numberphile
Catch David on the Numberphile podcast: https://youtu.be/9y1BGvnTyQA Professor David Eisenbud on the infamous Collatz Conjecture, a simple problem that mathematicians may not be "ready" to crack. More links & stuff in full description below ↓↓↓ Extra footage from this interview: https://y
From playlist David Eisenbud on Numberphile
The programming language created by John H. Conway
How can we program with fractions only ? The Fractran programming language invented by John H. Conway only consists of fractions, I explain how does it work in this video featuring the Collatz conjecture. --- Chapters: 0:00 Introduction 0:09 The Collatz conjecture 2:41 The Fractran progr
From playlist Summer of Math Exposition 2 videos
The hyperbolic Ax-Lindemann conjecture - Emmanuel Ullmo
Emmanuel Ullmo Université Paris-Sud February 7, 2014 The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort
From playlist Mathematics
Pyramidic Frustrums and the Collatz Conjecture
Modelling closed loops in the Collatz Conjecture with hyper-pyramidic bounds. Animated with Manim Community Fork: https://docs.manim.community/en/stable/index.html
From playlist Summer of Math Exposition Youtube Videos