Collatz Conjecture

The Collatz Conjecture

Introduction

The Collatz Conjecture is a conjecture made by Lothar Collatz, a mathematician who lived from 1910 to 1920. The conjecture is one of the most famous mathematical problems in history. To understand it, we must start with a random whole integer. Not a fraction. Lets say we use, 25. 25 is an odd number, the conjecture states if we use an odd number, we multiply by 3 and add 1. So 3x+1. This gives us 76. The conjecture states even numbers are divided by 2. So this is divided by 2 to become 38, which becomes 19, then 3x+1, 58, then divide by 2, 29, then 88, divide by 2, 44, divide by 2, 22, divide by 2, 11, 3x+1, 34, divide by 2, 17, then 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, and then 1 is odd, so 3x+1, we get 4. Now, the Collatz Conjecture states that with these 2 rules applied "If You chose any whole integer, and follow these 2 rules, that number will always fall into the loop of 4,2,1."

Understanding the Conjecture.

Now, lets define this more rigorously. An odd number is a number that is equivelant to 2y+1, where y is some number on a plane, it can be a complex plane, anything. However this conjecture deals with reals. An odd number must always equal 2y+1 with a specific whole y value, it cannot be fraction or decimal. Anyway, an even number is a number which is equal to 2y under the same conditions of a whole y value. Therefore, the conjecture can be defined as, (if x n=2y+1, x n+1 = (3(x n))+1, and if x n= 2y, x n+1 = x/2). We notice here that this problem seems simple, almost too simple.

The Problem

But it isnt. The problem with the conjecture is that it seems like on average, a number that is in this conjecture will always decrease, hence reaching an even number in the powers of 2. That is literally the only thing needed for a number to enter the 4-2-1 loop. The problem? Mathematicians havent been able to find a proof, and they also cant find a counter-proof, or a way to disprove the problem. And the lack of proof is from well, it being very hard. But, the lack of a counter-proof is not from lack of trying. The conjecture can be disproven in 2 ways, either, there is a specific seed number, as it is called, where the string after blows up to infinity, which has not been found yet, or there is a set of numbers that forms a loop, every number in this giant loop of numbers would not at all be related to any other number in the conjecture, as that would allow escape from said loop and hence a way to reach the 4-2-1 loop. But as it turns out, mathematicians have calculated that any loop besides 4-2-1, would have to be over 100 billion numbers long. And scientists have searched for a potential seed that leads to either of these methods to being true to the number 2^68. This number is around 295 quintillion. And they havent found, anything.

Current Progress

Mathematicians have been able to prove that the Collatz Conjecture follows very strict criteria. In 1976 mathematician Riho Terras proved that almost all Collatz sequences reach a value below their initial value. In 1979, the limit was reduced to almost all numbers from collatz sequences reaching a value below inital x, which we shall define as n, to the power of 0.869. Then In 1994, the limit was reduced again to n^0.7925. In the implied case, the term "Almost All numbers", has an actual meaning. It means as the numbers that you are looking at go to infinity, the fraction of numbers under this curve, the "Almost all numbers", approaches 1, or 100%. Then in 2019, our favorite mathematician, and one of the best currently alive ones, Terry Tao, Was able to show the Collatz Conjecture has even more strict criteria. As it turns out, almost all numbers from the collatz conjecture end up below any arbitrary function of f(n), so long as said function goes to infinity as n reaches infinity, a.k.a f(n)-->∞ as n-->∞. But the function can rise as slowly as you want. Unfortunately, this is still not a proof, or counter-proof.

Credits

Also special thanks to veritasium, I wouldnt have learnt about this problem without him.