#### Am Vor year

The Collatz Conjecture is the simplest math problem no one can solve - it is easy enough for almost anyone to understand but notoriously difficult to solve. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Prof. Alex Kontorovich for introducing us to this topic, filming the interview, and consulting on the script and earlier drafts of this video.

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References:

Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208. - ve42.co/Lagarias2006

Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963-1999). The ultimate challenge: the 3x, 1, 267-341. - ve42.co/Lagarias2003

Tao, T (2020). The Notorious Collatz Conjecture - ve42.co/Tao2020

A. Kontorovich and Y. Sinai, Structure Theorem for (d,g,h)-Maps, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.

A. Kontorovich and S. Miller Benford's Law, values of L-functions and the 3x+1 Problem, Acta Arithmetica 120 (2005), 269-297.

A. Kontorovich and J. Lagarias Stochastic Models for the 3x + 1 and 5x + 1 Problems, in "The Ultimate Challenge: The 3x+1 Problem," AMS 2010.

Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv preprint arXiv:1909.03562. - ve42.co/Tao2019

Conway, J. H. (1987). Fractran: A simple universal programming language for arithmetic. In Open problems in Communication and Computation (pp. 4-26). Springer, New York, NY. - ve42.co/Conway1987

The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1) [Computer software]. www.manim.community/

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Special thanks to Patreon supporters: Alvaro Naranjo, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

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Written by Derek Muller, Alex Kontorovich and Petr Lebedev

Animation by Iván Tello, Jonny Hyman, Jesús Enrique Rascón and Mike Radjabov

Filmed by Derek Muller and Emily Zhang

Edited by Derek Muller

SFX by Shaun Clifford

Additional video supplied by Getty Images

Produced by Derek Muller, Petr Lebedev and Emily Zhang

3d Coral by Vasilis Triantafyllou and Niklas Rosenstein - ve42.co/3DCoral

Coral visualisation by Algoritmarte - ve42.co/Coral

## KOMMENTARE

## Cosmic Nomad

I absolutely love how mathematicians always find the most random things to debate over!

Vor 7 Monate## ysdfdfk

It's not random. It just seems so for uneducated people.

Vor 11 Tage## Virtual Pilgrim

This video is just to embarrass mathematicians by providing evidence that they don't know very much. I looked on math from both sides now, from up and down, and still somehow, it's math's illusions I recall, I really don't know math at all.

Vor Monat## Renyx Ghoul & God Slayer

As it is chaotic in nature, stubborn and inquisitive = mathematicians

Vor 2 Monate## RicSouza

@Christ Loen the solution in itself is worth it. math is art, after all.

Vor 2 Monate## Diane Allen

I’d love a follow up on this that explores the callout sequences that don’t fall below their initial value. (Discussion started at ~12minutes with Riho Terras in 1976 discovering that most do).

Vor 5 Tage## FictTheCreator

I remember hearing about the Collatz Conjecture in high school. One of the ways I would while away my time in math class was inputting random numbers into the calculator and doing 3x+2 and x/2 until I got back down to 1, rest assured I never encountered any exceptions lol Blew my mind as a child and it's fascinating to see it as a window into the...almost fragility and beauty of math as a concept

Vor 20 Tage## Arvi Angelo

@bossa nova Ik theyre functions since 1 / 1 = 1 so yeah

Vor 10 Stunden## bossa nova

@Arvi Angelo I would recommend you learn the difference between a function and an equation. 3x+1 and x/2 are functions, you can't just treat them as equations

Vor 22 Stunden## Arvi Angelo

@LeVar Dotts right

Vor 2 Tage## LeVar Dotts

@Arvi Angelo I think I know what you meant to say there, but every integer is divisible by 1 since you will never get a remainder, including 0, i.e. 0/1=0. Now if you meant for 1 to be the numerator or the dividend, then in this situation it is only divisible by one number, which is 1.

Vor 2 Tage## LeVar Dotts

@Arvi Angelo Just to clarify, are you saying that the 3x+1 problem will always go back to 1 for any natural number, or are you saying the equation 3x+1=1 is true for any value of x?

Vor 3 Tage## Jason Laake Official

Also interesting to do a *Collatzception*: start with whatever number, count the steps and enter the number of steps as a new number. All the tests I did came to 5. And 5 needs 5 steps to get to 1. 16 leads to 4.

Vor 18 Tage## Infinity Plus 1

The more I know about math and science, the more I realize how much we don’t know

Vor Monat## shadyceddy

Fun fact: We are not mathematicians but we got interested by this.

Vor 10 Monate## D Evil

To be honest my question is WHAT IS THE PROBLEM? I am not a science student and I s*ck at it and my job has nothing to do with maths. What's shown JUST HAPPENS . So what we are supposed to SOLVE here ? If it happens then it happens , I just can't understand how this thing is a *problem* ? 😶🤕🥺

Vor 5 Tage## Arvi Angelo

and 3x is equal to 3 numbers added to each other and the 3 x's remain a mystery, it could be any number but So far, it is known that all numbers less than 20, eventually reach 11

Vor 17 Tage## Arvi Angelo

I think 3x+1 = 1 since 3x has no value?

Vor 17 Tage## Tara Lyn Nakamoto

Yeah

Vor 20 Tage## A1 AR1

I somehow get curious enough to open these videos but by halfway thru I’m so lost it’s a waste of time for me to keep going. 😵💫😵💫. Yes, I have a college degree, studied physics, algebra etc but never got not into math enuff to follow this stuff.

Vor 2 Monate## PsithyrosTV

I wonder if there's any correlation between the base number (x) and the number of steps it takes to get to the 4, 2, 1 loop

Vor Monat## Riemann's Last Theorem

I have seen this video many times, and I absolutely love it. Great Job Derek!❤🧡💛👍👍👍👍

Vor 15 Tage## Malek Abozraig

Your presentation is magnificent, well done 👍

Vor 12 Tage## Darsh khandelwal

The best video on youtube i have ever seen and amazed that for such a trivial looking problem, no one has yet been able to prove/disprove it.

Vor 2 Monate## Diego de Paula

Whoever created all those graph animations is an absolute master in after effects expressions

Vor 6 Monate## Josiah Brady

appreciate it

Vor 20 Tage## Rubén Ossorio

Logo l mmm yeah o lo

Vor 2 Monate## Abhishek Srivastava

It's made by Manim, an animation tool created by 3blue1brown

Vor 3 Monate## IRON MANUAL

EXACTLY WHAT I WAS THINKING

Vor 3 Monate## PotatoLord

I like men

Vor 3 Monate## Scyxurz

15:03 using 3x+1 in the negatives would be equivalent to using 3x-1 in positive. The same different loops would be found there too

Vor 10 Tage## Mike C

I remember trying my hand at this once and lost my train of thought because I was falling asleep at the time. Some equations are inherently unsolvable or unprovable by design.

Vor 26 Tage## Ruth Hopeful

12 million subs!! Congrats! It didnt prove Collatz conjecture but proves that after all, Math is one of the most interesting entertainments

Vor 23 Tage## Bailey Smith

the problem is that its not just 3x+1, its 3x+1, when x is even, x/2. you have a variable increasing x and a variable decreasing x. 3*x when X is odd will always give an odd product. +1 will make that product even. half of known numbers when divided by 2 are odd, and will eventually lead to multiple even numbers in a row, which will inevitably lower the number more and more

Vor 15 Tage## K. PacificNW

Everyone here: "...but just a maaaaybe I'll be the one to solve it."

Vor year## Shaun Hermans

@Cikong Li have you formalised this?

Vor 22 Tage## P. Jannat

😂😂😂😂😂😂

Vor 3 Monate## I wanted an emo png then this happend

XD

Vor 3 Monate## Rakheda Clarke

For real

Vor 3 Monate## vsink

@Milena Vision I did, I’m right, you’re wrong lol.

Vor 4 Monate## Tamás Kalocsai

Hi! Being proud as a hungarian as you mentioned Pólya, Erdős... and others who aren't mentioned here like Neumann, Teller, Kemény... just one thing about the pronounciation of hungarian names: Erdos is written with this letter: O-->Ő which sound like the french vowel "oe" in the word oeuf.

Vor 2 Monate## SirPikaPika

Honestly I feel like the negative numbers thing isn't that confusing It's the same as changing it from 3x+1 to 3x-1. 3x-1 has the same loops as negative numbers in 3x+1, just positive rather than negative

Vor Monat## MSTCYSAL

Interesting problem. As I am just a student, I can only prove that there wont be any loops except 4,2,1. only when 3x+1=(2^n)x it can form a loop. x=1/(2^n-3) and x, n can only be positive integers, so x can only be 1 (which means the only loop is 4,2,1). There is probably a way to prove whether is there any sequences to infinity but idk lol.

Vor 29 Tage## Neema Zadeh

> only when 3x+1=(2^n)x it can form a loop. The loop could be a long sequence of a's and b's, where a(x) = 3x+1 and b(x) = x/2. (If you'll allow exponents to represent repeated applications of the functions a and b) Your claim is basically saying that a loop must be of the form "(b^n * a)(x)", and -- you're right -- it seems to be the case that the only input that loops like this is x=1,n=2. But I don't see why a loop couldn't exist for some very large input x in any of the infinite number of forms: a^n_1 * b^m_1 * a^n_2 * b^m_2 * ... (where n_1, m_1, n_2, m_2, ... are arbitrary integers).

Vor 28 Tage## J Modified

"only when 3x+1=(2^n)x it can form a loop" This is only true if the pattern is an odd followed by some number of evens, and that is all you have proved - that there is only one loop in the positives that is an odd followed by some number of evens. Other patterns are possible.

Vor 28 Tage## Scott Nunnemaker

I mean if you want to recreate it with negative numbers you just switch it from “+1” to “-1” and you get the same result as the positive numbers. In the same way if you wanted to replicate what you do with the negative numbers use a “x3-1” for odds.

Vor 20 Tage## Agentkp

Mathematicians: Dont waste your time on this problem 20.7 million people: YES

Vor 10 Monate## Liam Wilson

@Frank Chary imagine if we do figure out warp technology and they actually call it warp technology lol

Vor 3 Monate## soobin doll

26million*

Vor 3 Monate## Abhijeet

Search about Abhijeetbyte Collatz Conjecture GitHub 😎🤣🤣👍👍..... Can't share Links on YouTube comments

Vor 5 Monate## Frank Chary

We need to work on practical problems that solve mankind's various problems such as air water and ground pollution, developing cleaner energy, food water and resource supply and more equitable distribution. Having solved these problems then we can move on to developing Warp technology to open up the final frontier.

Vor 7 Monate## Peaky2018

Thankyou for putting in the effort to make this video I actually found it really interesting

Vor Monat## Balam Jimenez

It was an amazing video, thank you. I am not a mathematician at all but I do have an observation/question/thought. Maybe the answer is not within the problem but outside the problem. This might just be babbling and outrageous so pardon my ignorance. Has anyone tried to develop the same problem for numbers in different bases? Like Sexagesimal numbers? The hypothesis would be that there is a number or series of numbers that are shared among Collatz conjectures of different bases. Of course it is unlikely all bases have the same Collatz conjecture model (3N +1 for odd numbers and divided by 2 for even numbers). I guess each base has its own Collatz conjecture model and that needs to be found first. So, after finding each base model, could be possible to find those numbers that are shared among conjectures. One or more of those shared numbers could be subject of a different loop than 4-2-1 in the base 10 Collatz conjecture. Probably the interaction of those numbers could create a unique loop different from each conjecture or a different unique tree that is not constrained by the base of the numbers used, and has its own behavior. What are your thoughts?

Vor Monat## Tomas

Well, the problem doesn't depend on the basis used, so it doesn't matter which basis we use. The only thing that matters is whether a number is odd or even

Vor Monat## There is a price to everything

Numerical gravity is honestly the best description for the pattern in this equation.

Vor Monat## The ruler of the internet

After watching this video i imagine a tree that grows for infinity. Every number is a branch or leaf that all connects back to the one seed that started the whole thing. Anyone else picturing something similar?

Vor 21 Tag## Marco Kapusta

This math problem is actually like my trading portfolio, I can start with any number but end at $ 1

Vor year## Not Sure

This is life, no matter how often you do something odd, you end up as one…

Vor 9 Tage## runciter naki

you posted the solution in your post there^ - hint; single character

Vor 2 Monate## DoYouLikeOculus

Then start at 0

Vor 6 Monate## Crasho

@Luca Disgusting Light Mode

Vor 6 Monate## Marco Kapusta

@Anonymous000 I love it when a plan comes together

Vor 6 Monate## Devanshi Singhal

What do you think happens to decimal numbers when followed the colatz conjecture?

Vor Monat## Akio

Positive integer*

Vor 29 Tage## CrayRT

Im not a mathematician, but Im now tempted to make a python script just so I can mess around with this.

Vor 14 Tage## alexander sze

Wow, how the hell did Brian Haselgrove get that?? I feel like that conjecture is quite cool to look into as well

Vor 23 Tage## Omgdodogamer

15:33 i dont know if anyone else noticed this but the amount of perfect squares in each number roughly equals the sqaure root of that number. sqrt(100) = 10, sqrt(1000) = 31.6, sqrt(10000) = 100

Vor Monat## TeamSprocket

@Uzaam Syed to do a proof by contradiction, you assume the opposite of what you're trying to prove, and then prove that it leads to an impossible result, thus your original assumption must be true

Vor 10 Tage## friendlyfireZA

There is definitely a geometric argument to made here. It's above me, but it's there.

Vor 19 Tage## Uzaam Syed

@Релёкс84 Never had to go past calculus 2 so it always puzzled me thinking about how a mathematician knows how to start "proving" something. we did some of it for geometry but that was like a decade ago lol. would you start by disproving it first?

Vor Monat## Релёкс84

@Omgdodogamer Yea you already said that. You formulated your conjecture, and I'm asking if you can prove it. I for one can, and it's a good exercice to try it yourself.

Vor Monat## Omgdodogamer

@Релёкс84 if you google perfect squares between 1 and 1 million it says 1000 and the square root of one million is 1000

Vor Monat## The Musical Stylings of Brent Bunn

Mad respect to the animators here. That must've been a lot of work.

Vor year## braum leon

i wonder if there an effective way to animate like this, or we just do it manually?

Vor 6 Monate## Pratana Kangsadal

Amen.

Vor 6 Monate## Häherfeder

How?

Vor 9 Monate## cisse de clercq

we do or best.

Vor year## johnsolo123456

Clearly this and problems like it, have the "encoding" of the structure of the integers in it, which is predominately the structure of prime numbers. The prime numbers have yet to be understood in a way which unravels their complexity, so it's no wonder this problem also escapes clarity. I wonder if you could make the layout of a "pathway" constricted with primes or something close and show how a "ball" like this problem would bounce through it.

Vor 25 Tage## tbcsuperman

11:00 imagine if the next number after that disproves the conjecture.

Vor 23 Tage## Rich Millich

I like to say that "mathematics is most often about the relationship between numbers and sets of numbers, not in computing anything." All too often we concentrate on utilitarian number crunching, which is boring, and why most students loathe math. Instead, patterns OF numbers and within sets of numbers are much more pleasing to the pattern making instincts of the human brain. Odd. Even. Positive. Negative. Fractions. Now here come the irrational numbers. Pi. Euler's Number. Phi, the Golden Ratio. Now here's a Cartesian plane. A unit circle. Sine waves, which we see and know in nature. Now here's a sphere. A hypercube. Fractals like the Mandelbrot Set. Procedures built with numbers that generate landscapes. Layers of infinity and an infinite number OF infinities, many of which we will never know. And somehow, within many of these mathematical constructs, we discover similar patterns over and over. Sometimes it feels as if there is a great unified system of patterns that revolve around each other, forming shapes, galaxies, anything we can imagine. THAT makes math beautiful. If we only taught it this way.

Vor 2 Monate## Michael Tritter

Pretty fascinating, but what really got me was 15:48 ... it looked like a pattern for primes numbers (or, actually, a pattern of patterns, but still discernable). I'm not a mathematician so I had to look it up to be sure, that we've yet to determine a predictable formula for primes, but visually, when you expanded from 1000 to 10,000 it looked, to me anyhow, that the movement of the primes could be predicted. Am I crazy or is there something there?

Vor 16 Tage## Michael Tritter

@spirou201 Doh. Thanks.

Vor 14 Tage## spirou201

It's not prime numbers, it's the numbers that are perfect squares. Or maybe I understood you wrong?

Vor 14 Tage## TheNordicNormie

Pretty much every subject in school is really interesting if I’m not forced to learn it

Vor 11 Monate## Guy

facts

Vor 6 Monate## Pratana Kangsadal

Amen.

Vor 6 Monate## Victor Kappel

You just described the main problem with the current education system on several places on the world: They don’t make you interested in learning the subject, they force it down your troath

Vor 9 Monate## How the turn tables

Yes

Vor 9 Monate## Joe Moore

Is it really surprising that negative numbers have different loops? It's changing the problem; you would end up with the same loops on the positive side if the problem was 3x-1.

Vor Monat## SaphiraSeraphina

Okay, there's a frequency graph on wikipedia. Could you approximate it with the chi²-distribution, they look somewhat similar.

Vor 14 Tage## Son Goku

Hi, can we try multiplying it by root? Sorry, i am just bad at maths. I need a simple explanation 👍

Vor 4 Tage## Richard Carter

I guess you would have to find an odd number that remains an odd number when you add one, because every positive number it seems will always end in 4,2,1. This equation is within everything depending on how you view it. From the first cell dividing into two and so on. Even the coral shape looked similar to DNA from above. It seems like an entirely new concept is what is being referred to when saying math isn't ripe enough. We may have to learn to plot certain points and imagine entirely new planes, like the rotating to certain angles to make the coral shape. That's ingenious thinking for sure.

Vor Monat## Rookie Lorenzo

@Hugo Delsing would u count infinity as even tho or odd is the case that could be made

Vor 17 Tage## blah blah

The very definition of an odd number is that it isn’t perfectly divisible by 2. So with any odd number to infinity, if you add 1, it’ll always be positive.

Vor 23 Tage## Richard Carter

@Boxingbros7890 yes, in the video he said that for some reason negative numbers act differently with the 3x+1 equation. I guess the different set of rules make it able to achieve different results. Also that wasn't the only loop, but he mentioned that there were some that will always end in other closed loops when dealing with negatives.

Vor 24 Tage## Hugo Delsing

There are no odd numbers that remain odd. All we need to check is 1 to 9 and that's it. Because whatever numbers you put in front, it is always an even number + (1 to 9). 58353561 is 58353560 + 1 and 6754938 = 6754930 + 8. And any even integer times any integer (in this case 3) is always an even number. And then we need to add (1 to 9) times 3, which will define if the final result will be odd or even. And 1 to 9 is already tested: 1 = 4 3 = 10 5 = 16 7 = 22 9 = 28

Vor 28 Tage## ParkieL

Oh my god, this poor animator. That is a serious amount of dedication. Looks fantastic!

Vor year## Kalliboy

@Mehtab Traveller Maybe I'll work with this someday Python fascinates me everyday

Vor 5 Monate## NotYayaNguyen

@Suppressed Nightmare how do u know it’s a “he”

Vor 5 Monate## Llama Man

@Lucky The Luckless Wolf I know I am

Vor 6 Monate## Lucky The Luckless Wolf

@Llama Man no, you're amazing

Vor 6 Monate## Suppressed Nightmare

He needs to be paid every single day $100,000 heh

Vor 6 Monate## Pancake

This kid one time went up to me and asked “are you good at math?” I said “yeah kinda” and he said “then what is 3x+1?” As a veritasium subscriber I knew he had no idea what he was talking about and just looked at the thimbnail 🤣

Vor 19 Tage## Surfer Dude

I've done a few proofs in my time. This problem seems to lend itself to proof by induction. But I gather that better mathematicians than I have already tried.

Vor 25 Tage## Teaspoon Tonn

If there is a loop available than with enough time you will hit it, especially on a downward trend pointing towards it, also loops are easier with lower numbers

Vor 29 Tage## Karoly VEGH

Now this was fascinating. Appreciate it.

Vor Monat## Pansoti

This sounds like a problem that we will one day show to a chaotic, but brilliant and creative child/teenager and he will just give us a counterexample in minutes and no one would know how

Vor 10 Monate## Lily Liao

@Zain Elsayed zain the incel

Vor 5 Monate## Jim Balter

@Stolfo Ch. No, Pags is the fool, not you.

Vor 6 Monate## Zain Elsayed

@IrokoSalei that’s not what i meant. im not saying that there arent any genius women out there.

Vor 7 Monate## coffee snek

I just realized that at 16:07 when the perfect square thing is being talked about, you can see parabolas in the grid that is shown. Call me crazy but this is cool

Vor Monat## Neema Zadeh

in a way, it's a plot of x^2 :)

Vor 28 Tage## Jake

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain

Vor 22 Tage## Vincent Boyle

this comment not getting the respect it deserves!

Vor 4 Tage## Johnathan Commodore

This theory looks similar to the way how a music spectrum analyzer operates on the graph as the music plays at different frequencies the light bars go up and down.All of the frequencies have a peak and fall flat to zero while all numbers representing the light bars hit their peak at one point during a song playing.

Vor 20 Tage## Blackwolf

Just an idea, when you are plotting the seed numbers, is there a pattern that repeats, like in fractals. If the pattern repeats, you have your answer. You heard it here first folks :D

Vor 5 Tage## Demens Clay

A big shoutout ot the graphics department for making this 100% more understandable!

Vor 3 Monate## Josiah Brady

no problem

Vor 20 Tage## gerkey

@myUserName thanks

Vor Monat## icebreaker900

TURN TO THE LORD JESUS CHRIST BEFORE ITS TOO LATE, GIVE YOUR LIFE TO HIM AND START WALKING IN OBEDIENCE, WITHSTANDING FROM ALL SIN AND WICKEDNESS, JESUS SAID THE PATH TO HEAVEN IS HARD AND NARROW, AND FEW FIND IT. MATTHEW 7:13-14, HEBREWS 5:9, JOHN 14:15, MATTHEW 7:21-26, 1ST CORINTHIANS 6:9-10, JOHN 3:16-21, JOHN 10:7-8, MATTHEW 10:26, AND LUKE 13:5. GOD BLESS YOU ALL.

Vor 2 Monate## Lance Dela Torre

Yet I still can't understand

Vor 2 Monate## Jdogg4089

@Gvko A big shout-out to you who couldn't be bothered to proofread!

Vor 2 Monate## Kemp Sanders

In this game, there’s more chance of hitting even than odd. Odd +50% even -50%. Even always follows odd. Odd or even can follow even. There’s also a chance of hitting a geometric sequence of evens (64, 32, 16, 8, 4, 2). I probably missed something important but that’s how I think about this. Rules of the game are rigged in favour of evens always winning (eventually) and then getting trapped in 4-1 loop. The loop is just because of the +1 I think.

Vor 18 Tage## Pandu

The interesting thing in this type of questions is that pattern. That's unique pattern makes the numbers unique.And more is we can understand these type of questions with this pattern.

Vor Monat## Axl

It’s easy: entropy applies to math, too. It‘s maths equivalent to organic life.

Vor 22 Tage## Emma

Interesting. I dont know if Im able to solve it, that is my wondering as well. Why say I can and afterwards the result wont be the right one.I ll watch tomorrow the rest of the video to see if my calculs fit in your response. Untill now 3x+1 shall be a linear function on xy ax...

Vor Monat## Walkastray

A couple of days ago he had a poll on what colour would evens and odds would be if they had a colour. The poll decided blue as even and red as odd. In this video, he has the evens as blues and the odds as reds. I love how much he cares about his community and the little details.

Vor year## Pratana Kangsadal

Amen.

Vor 6 Monate## Maria Maria

@Tyler Lawrence I do. My favorite number since I was a child was 7. When I learned to read I played a game in my head when I was little. I liked the words with odd letters because I would divide them in my head . Odd numbered words would have an even number on the left and right and an odd number in the middle. I liked to spell them backwards and speak backwards when I was bored. I liked it much better than the even numbered words.

Vor 6 Monate## IrokoSalei

Is a multiple of 2 : Kalm Is another number : REAL SH T

Vor 7 Monate## Uloola

@Jim Balter Odd is right.

Vor 8 Monate## Stuart McShitz

I stumbled onto something with the conjecture. Waiting on someone who's working on the program to run my equations. If I'm correct, 3x+1 will be followed by an infinite number of possible equations that will repeat the same phenomenon. Wayyyyyyy too much to do by hand.

Vor 2 Monate## Knowleaf of the Lions

I'm pretty sure it's always possible to get to one, because eventually, if you get an infinite number of chances for each number to get to one, than it's always possible.

Vor 18 Tage## Bobby Adams

I love math but am _'no'_ mathematician, given that, I'm curious, if the conjecture _'could'_ be solved, what's the point? Is there a tangible use that could result?

Vor 9 Stunden## nonyabusiness

We all got trolled by Collatz Dude said "Any number except 1 or 0" and literally had people busting their brains over it for decades

Vor Monat## Hanya manusia biasa.

Me : "That's interesting puzzle, maybe I can solve it" Me 22 minutes later : "oh."

Vor 10 Monate## The Rayven

@ItzCharmian fair enough...

Vor 10 Monate## Grzegorz Hordejuk

1 instead of 0 on the scale

Vor 10 Monate## The Rayven

@ItzCharmian which means you *HAVEN'T* solved it... If you had solved it, you would be more than willing to publish your findings, to prove you solved the equation... However, your lack of willingness to prove you solved the equation, only proves that the equation has not been solved because there is no proof that the equation has been solved... So in reality, the equation remains unsolved...

Vor 10 Monate## The music King

@ItzCharmian what is it

Vor 10 Monate## Anthony

The fact that even 0 ends ends up to the loop is already convincing me that this is impossible to solve

Vor Monat## Tomas

What? 0 remains 0

Vor Monat## Frank McCann

Perhaps imaginary numbers are Creative Intelligence s way of giving us the upper hand? Thanks 👍. Really good. They're more important than we ever thought.

Vor 8 Tage## Jerk Jerkington

The percentage of squares only goes down because you're counting in base ten, instead of "base square".

Vor 27 Tage## raxyls

just imagine if there was a full number that was neither odd or even.

Vor Monat## toe gang

I’m still trying to figure out what math problem we’re trying to solve.

Vor year## The Lazy Boston

@Jim Balter the Dunning Kruger effect is based on competence not stating facts the math equation makes it inevitable to get a bit wise number

Vor 8 Monate## The Lazy Boston

@Jim Balter I writing my thoughts now because it's the only way

Vor 8 Monate## Jim Balter

Lots of instances of the Dunning-Kruger effect in the comments here.

Vor 8 Monate## Sooraj

Fun fact: there is a closed loop for 5 if we subtract 1 instead of adding

Vor Monat## Elllie

this video was very interesting as numbers and math often do but reminds one of the last years at school I failed cause half of the video I was having trouble understanding but still cool none the less

Vor 4 Tage## Zen-oh Sama

I like that he said "7 great choice" cause most of the population, when they are asked to name a number they say 7. i said 5

Vor 5 Stunden## truth2belief

Just out of curiosity... Has anyone attempted to work on this problem by seeing if it's possible or impossible to get a prime number with the formula?

Vor 3 Tage## Devil The Feral

If the formula goes to a prime number, it's odd, since evens can't be prime as they are divisible with two, so one would add 3x+1 to that prime

Vor 3 Tage## Mr Scientific

Nice work Soviets. You got me.

Vor year## runciter naki

you fell into the 'trap' by yourself. Please take all the credit. What is a 'soviet'?, Propaganda is significant;y older than your avatar's face looks

Vor 2 Monate## Artificial intelligence plus lottery

Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Vor 7 Monate## Surya Prakash

@Ali Akram wwwwwwwwwwwwww¹wà

Vor 8 Monate## Martina Omeara

It's 4X the X doesn't mean multiply

Vor 8 Monate## Sameenoobgaming

Thehh uhhhh got me

Vor 8 Monate## Nathan Valentine

I am curious what happens If you reverse the steps? Of odd, divide by 2; if even multiply by 3 and add one.

Vor 6 Tage## Oh oi boi

My brain has melted into wonderful math soup... Yummy yummy in my tummy. Amazing video... if I stop and think about math. The idea of its existence. How it became to be or where it came from and the language of it... I think of the stars in the night sky or galaxies and the vastness of space and it's complexity. It is a miracle like he said and I enjoy 🤔 about all of it... Cause I'm going to find them numbers.

Vor Monat## Sarim Khan

wait but what if we try this formula on positive decimals(eg: 1.2954098765(i havent tried this myself))? what outcomes do we get?

Vor 21 Tag## Nick atnite

The reason 3x+1 is different using negative numbers is because when using negative numbers the + works like a - so it works like 3x-1

Vor Monat## Levy Chevy

My calculus professor just introduced this conjecture to us last week, and ever since then I've been shamelessly addicted to just bringing up a random number generator for a starting point and wasting away the hours.

Vor 10 Monate## Artificial intelligence plus lottery

Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Vor 7 Monate## DHRUV SHARMA

Same brooo

Vor 10 Monate## James Pronger

Blah blah blah more replys. "Think differently and simply"

Vor 10 Monate## Diego Castro

Nerd

Vor 10 Monate## Anime Fan Club

You have a great teacher if they motivated you to spend hours on this!

Vor 10 Monate## GlowstoneLove Pad

How about 3x-1? Edit: There are 2 loops: 1-2 and 5-14-7-20-10.

Vor 24 Tage## Morris Frayman

Maybe someone has made this observation (I can't go through 69034 comments), but if you keep taking log of log of log you should pretty quickly get to a number less than 1 in which case your next log is in negative territory. Then you can't take another log unless you're OK with complex numbers. So that's not a good formula to work with.

Vor 2 Monate## Sting Ray

What about using fractions? Do they still loop to 4 2 1?

Vor 15 Tage## Richard Groller

has anyone applied the contraction theorem to this problem?

Vor 2 Monate## Christopher

I'm not a mathematician but found this fascinating enough to watch the entire video.

Vor 9 Monate## Anshu Neupane

Same

Vor 5 Monate## Artificial intelligence plus lottery

Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Vor 7 Monate## Matthew

same lo

Vor 9 Monate## kpink gamer

Sameee

Vor 9 Monate## ThaKid Kane

What was the answer I don’t go time to watch the video but it should be 3 if it’s not it’s wrong

Vor 9 Monate## MagicCookie

If there is one number that creates a loop, or goes up to infinity, does that mean there are infinite ones that do so aswell?

Vor 22 Tage## eel

Yeah as the other guy said there would be an infinite amount of numbers leading to that loop by dividing by a power of two. But it doesn't mean that there are any other loops

Vor 19 Tage## Fhadi123

I think so, at least you can find infinitely many ones that leads you into the same loop or pattern going up to infinity. You could do this by multiplying the given seed by 2, 4, 8,16, or any integer power of 2. This new seed then goes through the halving process repeatedly until it goes back to the old seed and enters the loop or pattern. This is not too exciting of an answer, admittedly, might be wrong too. Your question is still intriguing, though, bc we can open the question to whether there are infinitely many distinct loops or diverging patterns.

Vor 20 Tage## Bon Sheedy

16:42 this blows my mind how in the 50's they can figure this stuff out.

Vor Monat## lumiwi

me: thinks of a possible solution also me: *perhaps i was the chosen one*

Vor Monat## Beytullah Dereli

I wonder what happens if you use numbers like 7.3 or 8.7 and using 3x+1.1

Vor Monat## Spookworm

I have never been someone who liked math during school, but for some reason I find it so completely interesting to learn about on my own time.

Vor 8 Monate## nowherenearby

@Spookworm nope

Vor 6 Monate## Spookworm

@nowherenearby You're entitled to your wrong opinion :)

Vor 6 Monate## nowherenearby

@Tza16 i dont think so, watching a surgeon do his work doesnt mean youre learning it (only if you already know a lot about medicine but whtvr)

Vor 6 Monate## Tza16

@nowherenearby it is for some people

Vor 6 Monate## nowherenearby

watching these types of vids isnt learning math but ok

Vor 6 Monate## Fred Lane

Although this may be of no value, it occurred to me that there is no upper limit of numbers, but there is a lower limit (the number 1). If there was no lower limit, the calculation would go in forever. The lower limit is the reason it stops.

Vor 2 Monate## Julien MARY

That channel is of public utility. Science is this: Questionning. At the time when science is perverted for political and unscrupulous reasons, such channel has the status of a lighthouse. Congrats dude. You do the job. -- Thanks. From France.

Vor 17 Tage## Michael Gohlke

Keep the rules the same Odd x3+1, even ÷2 Feed it negative numbers and some interesting stuff happens

Vor Monat## Rokaq

If every time you multiply an odd number by 3 and add 1, you end up with an even number, but not every time you divide by 2 an even number grants you an odd number, then chances are any number will eventually drop in value. The only possible result to the only operation that guarantees growth in the sequence is always going to be immediately followed by one or more consecutive decays. I don't think we need to prove or disprove the conjecture, we just have to learn to live with the fact that not everything around us owes us the gratification of absolute certainty.

Vor 17 Tage## Dark Ale

" we just have to learn to live with the fact that not everything around us owes us the gratification of absolute certainty" In math, it is often possible to in fact achieve the gratification of absolute certainty. There are many amazing problems that would have gone unsolved if we adopted this defeatist attitude!

Vor 16 Tage## DasSkelett

Your "one of the world's greatest living mathematicians" joke totally killed me.

Vor year## Artificial intelligence plus lottery

Found the Mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Vor 7 Monate## Karan

@dgcfgv vgb thanks

Vor year## dgcfgv vgb

@14k arena points what? I'm so confused lol

Vor year## Sarath Sreedhar

😂💯

Vor year## 14k arena points

Since Narak rythems with the english word ungramatmatically correct then that is correct.

Vor year## Colton Berry

One thought that crossed my mind... if 3n+1 is always even, then when you consider infinity, you only care about half of those numbers, because the other half will get cut in half anyway and fall into this bucket. What's also interesting is that the numbers you get when you have a series of divisions by 2, which you have no choice but to get, is that the distance between that even number and its closest power of 2 decreases. We know that if the number reaches a power of 2, it reaches 1, so this essentially tries to say that the function will eventually convert any number to a power of 2 and send it down the line to 1.

Vor 24 Tage## jursamaj

@jfitz458 "A pretty good idea" is not "I 100% know". And neither is a mathematical proof. Having a pretty good idea is where you *start* doing math, not where you stop. That is "the point of proving it": to actually prove it.

Vor 9 Tage## jfitz458

@jursamaj I don't 'know', but I've got a pretty good idea.

Vor 9 Tage## jursamaj

@Zato Zastovic You *think* you know it. Since it hasn't been proven, you don't actually know it.

Vor 21 Tag## Zato Zastovic

@Colton Berry Thi is what I wanted to say. I 100% know all numbers are gonna get to 1. What is the point of proving it?

Vor 21 Tag## Colton Berry

@Jaiden Carmona I think for this problem, we are only using integers. Decimals have their own thing going on with infinite decimal expansion.

Vor 22 Tage## Chaz titan

Isn't 3n a constant which can be delete and you always end with 1 in calculus? Eventually u walk away with a number ending in xxxxxxxxx0, no matter that start number

Vor 28 Tage## עדן פיבק

Thanks, very nicely explained and produced. Question: if the problem we are trying to solve is "Whether or not a number will hit infinity, don't we have to define what exactly is infinity? I mean why can't a number go back down from infinity when divided by two? is infinity is also a closed loop? (( I'm not a mathematician so be gentle on me (: ))

Vor 17 Tage## J Modified

Generally, you would prove that for any element of a specific Collatz sequence, there is guaranteed to be a following element that is higher. To take a much simpler example: x->x+1 clearly "goes to infinity" and it is trivial to show this - for any element of the sequence the next element is higher.

Vor 16 Tage## softed

Fun fact: If we find just one of these numbers, that won't eventually end up at 1 and loop, we would find an infinite amount of same numbers.

Vor Monat## softed

@Chodxniel Mobile Bruh, if we find just one of that numbers that don't loop, they would be divided by 2 or multiplied by 3 with the addition of 1, then it would mean that the number we got also can't loop because it's the result of the previous number that doesn't loop.

Vor Monat## Chodxniel Mobile

@softed why probably? By what logic did u arrive at that conclusion?

Vor Monat## softed

@Chodxniel Mobile Um, probably yes.

Vor Monat## Chodxniel Mobile

Um, no

Vor Monat## Tomas

Not necessarily, they could still form a loop

Vor Monat## Javier Solis

The animation is everything here.

Vor 10 Monate## Artificial intelligence plus lottery

Found the Mathematical Phenomenon A VERY interesting channel - " Artificial Intelligence plus lottery"

Vor 7 Monate## Evan Berger

I wouldn't have understood it otherwise

Vor 10 Monate## everywhereman900

@DON'T IM GONNA SAY THE N WORD

Vor 10 Monate## milkshake

@DStake i didnt even read the pfp the text is too small to see LMAO

Vor 10 Monate## DStake

You both just fell for his trap lmfao

Vor 10 Monate## Sultan Of Mad Olives

Have y’all done an episode on Fermat’s Last Theorem? I think that was finally proven in 1992. It’s like A^n+B^n=C^n for all natural numbers n…or something.

Vor 14 Tage## Mathematics Man

I don't know whether i able to solve this or not but still I invest time in it because no one knows whose hard helps other to discover it

Vor Monat## RedSchmit

15:03 You said "Why should there be disconnected loops on the negative side of the number line, but not on the positive side?". 3x+1 moves away from 0 on the positive side, but toward 0 on the negative side. Applying 3x+1 to the negative side is like doing 3x-1 to the positive side.

Vor 15 Tage## J Modified

That explains why they are different, but not why 3x-1 should have multiple loops while 3x+1 does not.

Vor 15 Tage## Omri Abraham

You will get loop numbers with any odd number you will add (3x+1)/2 or (3x+3)/2 or (3x+5)/2 and on and on

Vor 10 Tage