There exists an exciting conjecture that is extremely easy to solve and so interesting that if I wrote its statement here, even you would be tempted to grab a pen and paper to give it a try. And what if I say that despite this, no one has ever been able to prove it true. Well, that’s the beauty of the Collatz conjecture.

Going by the dictionary definition, a conjecture is an opinion or conclusion based on incomplete information. Several conjectures are still awaiting concrete proof in mathematics, and the Collatz conjecture is one of them.

Lothar Collatz: The father of Collatz conjecture

Although most of you have heard his name for the first time today, Lothar Collatz was a substantial mathematician who made a mark for himself in the twentieth century. He was a personality who was convinced that mathematics and mathematicians had a responsibility to apply their results to and be motivated by real-world phenomena. Eventually, he came up with one of the fascinating conjectures in the history of mathematics. Precisely speaking, Lothar Collatz introduced the idea of Collatz conjecture in 1937, almost two years after receiving his doctorate.

Lothar Collatz: The brain behind the Collatz conjecture
Lothar Collatz | Image: MacTutor

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Statement of Collatz conjecture

Collatz conjecture primarily concerns sequences starting with any positive integer. It says that if we start with any positive, say n, then 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, that is n/2.
  • If the previous term is odd, the next term is 3 times the previous term plus 1, that is 3n+1.
Collatz Conjecture: The Simplest Conjecture in Mathematics That No One Has Ever Proved 1

The series can be constructed by applying these two rules again and again. The conjecture says that no matter what the value of n is, the sequence will always reach 1. Don’t believe me? Let’s take an example. Let’s start with 10. So following the steps mentioned above, the sequence would be 10, 5, 16, 8, 4, 2, 1. This one was simple. Let’s take another example, say n = 50. So the sequence would become 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. I know you are now tempted to check it for several other integers. So what are you waiting for? Just grab a pen and paper!

The Collatz conjecture is also sometimes known as the 3n + 1 problem, the 3n + 1 conjecture, the Ulam conjecture, Kakutani’s problem, the Thwaites conjecture, Hasse’s algorithm, or even the Syracuse problem. Moreover, the sequence of numbers involved in the series derived from this conjecture is sometimes referred to as the hailstone sequence or hailstone numbers or as wondrous numbers. This is because the numbers in a Collatz orbit go up and down, just like how hailstones behave in the clouds where they form.

Collatz Conjecture: The Simplest Conjecture in Mathematics That No One Has Ever Proved 2
Binary tree showing Collatz Conjecture | Image: Semantic Scholar

So simple, yet no proof?

Now looking at the statement of Collatz conjecture, you might be wondering that it’s so simple that anyone can prove it easily. But this is not the case! For decades, the proof of this conjecture has appeared as a hard question for mathematicians all across the globe. To date, extensive titrations have been performed for an unmatchable set of numbers, going up beyond the 20th power of 10, beyond two hundred quintillions. Even every single one of them has produced a sequence resulting in 1.

I know. It’s a pretty good number to assume that Collatz conjecture holds for every positive integer. But this doesn’t apply to the world of mathematicians. Even after having proved this conjecture true for so many numbers, a single counter-example (if found) would be enough to establish that the conjecture is wrong.

So despite the quintillion of positive examples on the conjecture’s side, we need logical proof to demonstrate the authenticity of the Collatz conjecture. And this logical proof is something that we don’t have yet!

However, with the advancements taking place, numerous brains are developing rational proof for this conjecture. Over the years, several positive developments have been seen, and it is expected to have access to concrete proof in support of Collatz conjecture in the near future.

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Rishav Roy Chowdhury

Great effort to share such a simple yet powerful conjecture in maths. Keep it up!

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