## Leonhard Euler

## Collatz Conjecture

###### Ceyda Toprak

Do you like solving problems? Have you ever considered being a mathematician because one of the most essential parts of mathematicians’ lives is solving problems? There are various types of problems. Some of them require the help of supercomputers, and some require mathematicians to spend their entire lives trying to solve them. Also, there are a few others that seem unsolvable, despite the general belief that all math problems should eventually be solved (Shields).

##### The History of the Unsolved Math Problem

The Collatz conjecture, or the “3n+1 problem,” hasn’t been solved yet. The Collatz conjecture was introduced by a German mathematician called Lothar Collatz in 1937. It seems like an easy, straightforward question, but it is actually the opposite. The conjecture suggests that “if you repeat two simple arithmetic operations, you will eventually end up transforming every positive integer into the number one” (Shields). However, there is a problem because whether it is correct for all integers is unknown. There is still a possibility that with some number, the sequence goes into infinity.

Even though millions of natural numbers have been tested by mathematicians, and it hasn’t been proven wrong by anyone, no one has proved the conjecture unconditionally correct, either. A Hungarian mathematician called Paul Erdos said, “Mathematics may not be ready for such problems" (Shields).

This conjecture was put forward by Collatz two years after he received his doctorate at the University of Berlin. Even though all the calculations suggest that the conjecture is correct, it has remained unsolved for 86 years.

##### Why Is the Collatz Conjecture Also Called the '3n + 1' Sequence?

The conjecture of Collatz is also called the “3n+1” sequence because for it to be generated, a positive number is picked and two simple rules are followed: “If it’s even, divide it by two, and if it’s odd, triple it and add one” (Shields). Therefore, “3n+1.” Despite the starting number, if you follow those two rules as many times as necessary, you will reach number one eventually.

For instance, start with the number nine. It's an odd number, so you insert nine in place of n in the 3n+1, which equals 28. That's an even number, which means you need to divide it by two, which gives us 14. Here is the rest of the sequence:

14 / 2 = 7

7 x 3 = 21 + 1 = 22

22 / 2 = 11

11 x 3 = 33 + 1 = 34

34 / 2 = 17

17 x 3 = 51 + 1 = 52

52 / 2 = 26

26 / 2 = 13

13 x 3 = 39 + 1 = 40

40 / 2 = 20

20 / 2 = 10

10 / 2 = 5

5 x 3 = 15 + 1 = 16

16 / 2 = 8

8 / 2 = 4

4 / 2 = 2

2 / 2 = 1

Therefore, if you start with the number nine, the Collatz sequence is 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. “If you do it again from the number one, an odd number, you multiply by three and add one. From there you get four, which quickly reduces back to one. This begins the loop that never ends” (Shields).

##### Limited Breakthroughs With the 'Hailstone Sequence'

The numbers that are generated in the Collatz conjecture have another name, which is the “hailstone sequence.” It can be seen from the sequence listed above that “the numbers go up and down and up and down like hailstones in a storm cloud, being lofted up, collecting ice and, after falling into a lower part of the cloud, blown upward again. At some point, they plummet to the ground” (Shields).

Some certain numbers, which fall quickly once you reach them in the calculations are present, but they all fall to one eventually. Therefore, the Collatz conjecture works for millions of numbers — any number with less than 19 digits, in case you are planning on testing it yourselves — but mathematicians try to understand why. If they found out the reason, they could say with certainty that it works on all natural numbers (Shields).

The Collatz conjecture involves an infinite number of integers. Even the most powerful supercomputer is not able to check every single number to see if the conjecture holds true.

There is a mathematician who has made a bit of a breakthrough in the Collatz conjecture in recent years. Terence Tao published a paper with the title “Almost All Collatz Orbits Attain Almost Bounded Values” in 2019. Tao earned his Ph.D. from Princeton when he was 21 years old and became the youngest-ever math professor at UCLA when he was 24. Moreover, he won the Fields Medal at 31 years old. However, his big news about his Collatz breakthrough has two “almosts” in it.

Basically, the results of Tao point to a new method to approach the problem and note how rare it would be for a number to diverge from the Collatz rule. It could be rare, but not necessarily nonexistent (Shields).

As a result, that is the closest anybody could come in recent years to solve the Collatz conjecture. If you want to solve it yourself, don’t forget to start with numbers with more than 20 digits.

References

Shields, Jesslyn. “Even the Smartest Mathematicians Can't Solve the Collatz Conjecture.” HowStuffWorks Science, HowStuffWorks, 15 Feb. 2023, https://science.howstuffworks.com/math-concepts/collatz-conjecture.htm.

Figure References

“The Collatz Tree.” Algoritmarte, 2020, https://www.algoritmarte.com/the-collatz-tree/. Accessed 2023.