Collatz Conjecture Calculator

Simple things are sometimes surprising: explore a problem that makes mathematicians say, "I'm not going to do that" with our Collatz conjecture calculator!
Here you will learn:

• What the Collatz conjecture is;
• Why it is driving mathematicians crazy;
• How to calculate the Collatz (or hailstone) sequences; and
• How to use our Collatz conjecture calculator.

Ready to learn more about this mathematical mystery? Just be careful — trying to break the conjecture is dangerous. Remember to stop after a while!

A "conjecture" is mathematics lingo for something we're pretty sure is true, but we can't find a way to prove it. Quite frustrating, probably! The Collatz sequence is formed by starting at a given integer number and continually:

• Dividing the previous number by 2 if it's even; or
• Multiplying the previous number by 3 and adding 1 if it's odd.

The Collatz conjecture states that this sequence eventually reaches the value 1.

It is wonderfully simple, and yet every initial number ever tried returned 1 sooner or later — but no one has been able to prove it in almost a century!

The rules of the Collatz conjecture are, formally speaking:

We use the modulus operation (visit our modulo calculator to learn how to use it properly) to distinguish odd and even numbers. The second equation explains why the Collatz conjecture is sometimes also called the 3x+1 problem.

Starting with any number, how will the sequence behave? Will it go up to infinity, or down to zero? Will it loop forever?

Choose an initial number first. Let's say 11. The sequence goes up and down:

See the end? Once it reaches 4, it collapses into the loop $4, 2, 1, 4, 2, 1$ which it will never escape!

Choose any other number, and eventually, the sequence will end up in the $4, 2, 1$ loop. The oscillations may be big, as for $x_1 = 31$ which reaches a peak of more than 9,000 before falling down to 1.

This wildly unpredictably oscillating behavior earned these sequences the name "hailstone sequences", because the path of the sequence resembles that of a hailstone in a cloud before reaching the ground: swinging up and down before falling (preferably not on our heads).

The sequences follow a random pattern, and by just looking at the initial term, it's impossible to say how its sequence will behave without computing the next steps: mathematicians say that it is an undecidable problem. In layman's terms, there's no computer program that can take a number and say if it will or will not reach one.

This is why the 3x+1 problem is such a problem for mathematicians: at the moment, the only thing we can do is to brute force our way through numbers, trying to find one of them that would escape the $4, 2, 1$ loop. And things don't look promising: researchers have tried numbers up to (are you ready?) $295,147,905,179,352,825,856$ — more seconds than have passed since the Big Bang! And not a single number didn't end up at 1.

And in an attempt to show the Collatz conjecture who's boss, researchers tried the number $2^{100,000}- 1$ (the minus one was used to give the final flex, apparently). That number is 30,000 digits long, and guess what? After almost a million and a half steps, it ended up at 1.

The search for a counterexample (one that doesn't end up in the $4, 2, 1$ loop) is still undergoing, but many mathematicians think that this problem is out of the reach of our knowledge at this time.

The Collatz conjecture and its hailstone sequences are good examples of chaotic behaviour. If you need to balance it with some order, try out or Fibonacci sequence calculator or golden ratio calculator!

All you have to do is choose a number! Our Collatz conjecture calculator will show you:

• A chart of the sequence;
• The stopping time (the number of steps before reaching 1 for the first time); and
• A table with all terms of the sequence.

Did you try a negative number to see if you could break the tool? We covered that too, but mathematicians are even more worried about it: there are three loops (starting at $-1$, $-5$, and $-17$) for negative integers, and no-one knows why they exist!

You can also decide to edit the default rules, but we don't guarantee that something interesting will come out of it. Try anyway — maybe the next mathematical nightmare will be named after you!

Now that you know what the Collatz conjecture is, you can also understand that it is unlikely you are going to use it in real life. There are more useful (but maybe less thought-provoking) sequences in math: we have some calculators for some of the more "conventional" of them:

• Arithmetic sequence calculator;
• Geometric sequence calculator.