What is collatz conjecture: Definition and 18 Discussions

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 the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the 3n + 1 problem (or conjecture), the 3x + 1 problem (or 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." Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".

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  1. E

    Collatz conjecture brought me here

    These are the two tables I developed trying to untangle how the Collatz Conjecture operates, and this certainly helps clear the water a bit
  2. M

    I Breakthrough in 3n+1 Problem: Collatz Conjecture

    I did a study of the Collatz conjecture and found that all even numbers can be removed from the Collatz conjecture because even numbers act as the connecting links between odd numbers. What do you think about it? Is it a breakthrough in 3n+1 problem? 27 -> 41 -> 31 -> 47 -> 71 -> 107 -> 161 ->...
  3. elcaro

    I Is this a proof of the Collatz Conjecture?

    Note as soon as the term 3N+1 become divisible by a power of 2 we can repeatedly divide by 2. For the proof below we rearrange the sequence so it becomes: First step: If N is odd, multiply by 3 and add 1. Each next step: - Repeatedly divide by 2, as many times as the number k, which is...
  4. Chris Miller

    B Exploring Variations of the Collatz Conjecture: A Computational Approach

    The Collatz problem is perhaps the only unsolved math problem I actually understand. It "feels" like a proof would be trivial, though obviously it isn't. Been playing with different variations in hopes of understanding it better. Is it a set problem (proving there's no intersection between two...
  5. I

    A Can this proposition be proved in the Collatz conjecture?

    Can this proposition be proved and become a lemma in the proof of Collatz conjecture? $$collatz(n) \geq \lfloor \frac{log(n)}{log(2)} \rfloor.$$
  6. J

    Exploring the Collatz Conjecture: 2000 to Now

    I have been interested in the attempts to prove this Conjecture since 2000 and like many others (eg Ken Conrow) I have tried to find a convincing solution. Today I read on this forum what looks like a proof that there cannot be an internal cycle beyond 4:2:1 but I don't think the author realizes...
  7. Dtriction

    I Can this method be used to prove the Collatz Conjecture?

    There is a graph showing n on its x-axis and its total stopping time on its y axis. From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to...
  8. F

    Is the Collatz Conjecture's Behavior Predicted by Binary Representations?

    From what I can tell off of Wikipedia and Wolfram, it doesn't look like this is currently known. Regrettably, I live in a social vacuum of mathematical pursuits, so I've come here in the hopes that someone can tell me if this is really new information or simply a retread. Brief Collatz...
  9. F

    Method for proving the collatz conjecture, would this work?

    Would it be possible to prove the collatz conjecture indirectly by demonstrating rules that apply to 'Collatz-like' conjectures? (I call anything where you simply change the values in the 3n+1 part of the conjecture to other values, holding everything else the same a Collatz-like conjecture)...
  10. R

    Where's the flaw in this proof of the Collatz Conjecture?

    The conjecture states that: Given a positive integer n, If n is even then divide by 2. If n is odd then multiply by 3 and add 1 Conjecture: by repeating these operations you will eventually reach 1. Proof: Let n be the smallest positive integer that is a counterexample...
  11. V

    A Conjecture on the Collatz Conjecture

    I have created a program in javascript that has tested integers on the collatz conjecture. Recall that the collatz conjecture says given any natural number n you must divide n by 2 if it is divisible by 2 and multiply n by 3 and add 1 if it is not divisible by 2. Repeat this process and you...
  12. C

    Collatz Conjecture: Plotting Integer Steps with Joe

    Hello there everyone! I've written a lovely little program to go through the tedious process of testing numbers in the "If odd 3n+1; If even n/2; Repeat." scenario. It then saves all of the numbers, starting with the integer being tested, and ending with "1" in a file of the form...
  13. P

    Collatz Conjecture - Bouncing Ideas

    Hey, this is my first post, so... Hello everybody! I've been looking into the Collatz conjecture, and like most mathematically minded people, been completely absorbed by it. I'm looking to bounce some ideas off other people, kind of a peer review of a couple things if you will. I'll be the...
  14. M

    Has anyone read Ken Conrow's Collatz Conjecture website?

    http://www-personal.ksu.edu/~kconrow/" I mean really read it, to where they understand it, not just to the point where their eyes glaze over? I see his site cited quite frequently, but I don't know if I've ever seen any critique. I admit, I was intimiitaded by his site at first. But...
  15. P

    Collatz Conjecture Paper: Check it Out & Give Opinion

    The paper at this site "http://uts.awardspace.info" looked interesting to me, but would anyone else familiar with this problem, it's been around since the 30's, check it out and give an opinion.
  16. H

    Logical Systems+new Principles+attempt To Solve Collatz Conjecture

    LOGICAL SYSTEMS+NEW PRINCIPLES+ATTEMPT TO SOLVE COLLATZ CONJECTURE. First i would like to say that am honoured to share my thought with great people in here who always provide help.I will not say iam right or wrong.I hope this post will be aspark to good mindes. I have viewed the laws of...
  17. H

    Logical System+new Principles+attempt To Solve Collatz Conjecture

    LOGICAL SYSTEMS+NEW PRINCIPLES+ATTEMPT TO SOLVE COLLATZ CONJECTURE. First i would like to say that am honoured to share my thought with great people in here who always provide help.I will not say iam right or wrong.I hope this post will be aspark to good mindes. I have viewed the laws of...
  18. F

    Is Tripling Odd Numbers Necessary in the Collatz Conjecture?

    Hey, i was reading about the Collatz conjecture, where, if you take a integer, divide it by 2 if its even and triple it then add one if it's odd, and do it over and over again, the result would be one. I was thinking, "wouldnt it have the same effect if you didnt triple odd numbers?" am i wrong?