The Collatz Conjecture states that for any positive integer n, the Collatz sequence will eventually reach 1.

The Collatz sequence is:

So if n is even, divide it by two, and if n is odd, multiply it by three and add one.

So, for example, 36, 18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

It is still unknown if every positive integer will have a Collatz sequence that leads to 1.

A008884 is the Collatz sequence for 27, taking 111 steps.

The number below 10^18 that has the highest step count is 931386509544713451, which takes 2283 steps. Challenge: write all the steps and send me an image.

The only known loop in the Collatz sequence, or a cycle, is (1,2). But if we reach 1, it’s over.

Terence Tao created a huge leap in our understanding of the Collatz Conjecture. The article about it is here.

Maybe tomorrow, I will post a detailed version of this post.