This page uses JavaScript. The fact that you are seeing this information indicates that you lack JavaScript, or that you have it but it is disabled. You are welcome to remain on this page but, without JavaScript, you will miss virtually all that it has to offer.

Tao takes us to the Collatz Conjecture. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go.

Take any natural number. There is a rule, or function, which we apply to that number, to get the next number. We then apply that rule iteratively, and see where it takes us. The rule is this: If the number is even, then divide it by 2; if the number is odd, then multiply by 3 and add 1. For example, consider 10. It’s even, so the rule says to divide by 2, taking us to 5. Now that’s odd, so we multiply 5 by 3 and then add 1, landing us on 16. Now 16 is even, so we cut it in half to get 8. Even again, so halving gets us 4. Now 4 is even, so we take half, getting 2, which is even, and cuts in half to 1.

Start with numbers other than 10, and you’ll still inevitably end at 1 … we think. That’s the Collatz Conjecture.

It’s definitely true for all numbers with less than 19 digits, so that covers whatever you probably had in mind. But even if computers check up to 100 or 1,000 digits, that’s far from a proof for all natural numbers.

The number of iterations it takes to first reach 1 is the “stopping time.” In the table below, we just call it number of "Stops".

Binary search tutorial: https://www.geeksforgeeks.org/binary-search-in-javascript/

function binarySearch(sortedArray, key) {
var start = 0;
var end = sortedArray.length - 1;
while (start <= end) {
var middle = Math.floor((start + end) / 2);
if (sortedArray[middle] === key) { // found the key
return middle;
}
else if (sortedArray[middle] < key) {
start = middle + 1; // continue searching to the right
}
else { end = middle - 1; // continue searching to the left
}

// key wasn't found

return -1;

}


Collatz Calculations
 
Argument Stops Explanation
1 0 1
2 1 1
3 7 10 5 16 8 4 2 1
4 2 2 1
5 5 16 8 4 2 1
6 10 3 10 5 16 8 4 2 1
7 16 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
19 20 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
21 7 64 32 16 8 4 2 1
23 15 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1





 Overview  |  Allan Marain  |  Norman Epting, Jr.  |  The Practice  |  Selecting a Lawyer 
 Trial Transcript  |  Jobs and Internships 

 Abandoned Cars  |  Aiding Suicide  |  Animal Cruelty  |  Appeals  |  Arson  |  Assault  |  Bad Checks  |  Bribery  |  Burglary 
 CDS in Vehicle  |  Cell Phones  |  Child Pornography  |  Cocaine  |  Computer Crimes 
 Conditional Dismissal  |  Conspiracy  |  Credit Card Fraud  |  Drug Crimes  |  Drug Deaths  |  DWI/DUI 
 Eluding  |  Endangering  |  Entrapment  |  Evidence  |  Expungements  |  False Swearing 
 Federal Appeals  |  Federal Crimes  |  Fictitious Reports  |  Forgery  |  Graves Act  |  Guilty Pleas  |  Gun Law  |  Harassment 
 Hazing  |  Homicide  |  Identity Theft  |  Insurance Fraud  |  Invasion of Privacy  |  Jury Service  |  Juvenile Delinquency 
 Kidnapping  |  Leaving the Scene  |  Lewdness  |  Mail Fraud  |  Manslaughter  |  Marijuana  |  Megan's Law 
 Miranda Warnings  |  Money Laundering  |  Municipal Court  |  Murder  |  NERA  |  No Insurance  |  Official Misconduct 
 PCR  |  Peeping Tom  |  Perjury  |  Point Assessments  |  Police Misconduct  |  Possession  |  Probation  |  Prostitution 
 PTI  |  Receiving Stolen P'ty  |  Red Light Cameras  |  Refusals  |  Resisting Arrest  |  Revoked List  |  Robbery 
 Sexual Assault  |  Shoplifting  |  Stalking  |  Street Crimes  |  Tax Crimes--Federal  |  Tax Crimes--NJ 
 Terroristic Threats  |  Theft  |  Theft by Deception  |  Theft of Services  |  Traffic Offenses 
 Unauth'd Prac Law  |  Underage Drinking  |  Underage Gambling  |  Urinating in Public 
 Vehicular Homicide  |  Violation of Probation  |  Weapons Possession  |  White Collar Crimes  |  Wire Fraud 

 Discrimination  |  Domestic Violence  |  Drivers Licenses  |  Excessive Force  |  Freedom of Speech 
 Personal Injury  |  Whistleblowers 

 Case Review  |  Super Links  |  No Exit  |  Directions  |  Poker Portal  |  How Did We Do?  |  Collatz Conjecture  |  PGP Public Key 

 The Bottom Line