Collatz Conjecture

The Collatz Conjecture is one of the most famous mathematical problems in history, made by Lothar Collatz, a mathematician who lived from 1910 to 1920.

Overview

The Collatz Conjecture states that if a whole integer $x$ is odd, we let $3x + 1$, whereas if $x$ is even, we let $\frac{x}{2}$. For example, 7 is odd, so we let $7(3)+1=22$; 22 is odd, so we let ${\frac {22}{2}}=11$. From here, we get these next numbers: 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1. From any whole integer, the process will always result at the 1 → 4 → 2 → 1 loop.

An odd number is a number that is equivalent to $2y+1$, where y is an integer and a number on a plane, complex plane, etc. However, an odd number must always equal to $2y+1$ with a specific whole $y$ value. An even number is a number which equals to $2y$ under the same conditions of a whole $y$ value. Therefore, the conjecture can be defined as: if $xn=2y+1$ then $xn+1=(3(xn))+1$, and if $xn=2y$ then $xn+1={\frac {x}{2}}$.

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.