The Collatz Conjecture

One of the most difficult unsolved problems in mathematics is, in principle, so simple it could probably be easily understood by a 10-year-old.

The Collatz conjecture is known by several different names, including the Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, and the Syracuse Problem. However, it was most likely first posed by Lothar Collatz in the 1930s, and since then it has wasted the time of numerous mathematicians.

The algorithm itself is straightforward to understand:

• Start with any number, 𝑛

• If 𝑛 is even, halve it: 𝑛 → 0.5𝑛

• If 𝑛 is odd, multiply it by three and add one: 𝑛 → 3𝑛 + 1

An example of a run-through of this algorithm, beginning with 𝑛 = 6 would be:

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → 1 ...

You’ll notice that the sequence becomes stuck in a loop of 4 → 2 → 1 → 4 ... This is the premise of the Collatz conjecture - no matter the starting value of 𝑛, we will eventually reach 1.

The numbers in Collatz sequences are sometimes referred to as hailstone numbers. Collatz sequences can quickly rise and fall, often settling at a low number before suddenly shooting back up to a much higher value, but eventually, all of them fall back down to 1. This mirrors the motion of hailstones in a storm cloud, which can bounce around for a long time before eventually falling to the ground.

Something to remember is that the Collatz conjecture is exactly that - a conjecture. This means that so far, nobody has been able to prove it. Despite this, there has been some important work done on the problem. Mathematicians have tested the Collatz conjecture for every integer up to 2⁶⁸ - almost three hundred quintillion. Every single one ends up reaching 1, but this alone cannot prove the conjecture.

More recently, in 2019, Terence Tao, one of the greatest living mathematicians, managed to prove that the Collatz conjecture is ‘almost’ true for ‘almost’ all numbers. His work built off of research performed in the 1970s, which shows that Collatz numbers ‘almost’ always reach a number that is smaller than the starting point. Tao extended this research using a method of statistically analyzing the long-term behavior of partial differential equations; he eventually managed to show that at least 99% of numbers greater than one quadrillion will eventually reach a value below two hundred.

Still, not even Tao could formulate a full proof of the Collatz conjecture, which poses the question of whether it is even true. After all, it is possible there is a number that will continue to grow towards infinity, never reaching 1. Equally likely is the possibility that there is a closed loop of numbers, where the values will cycle and never reach 1.

New mathematicians are warned to stay away from the Collatz conjecture; many have spent years of their careers working on futile attempts to prove it. But at what point should we start looking, not to prove it, but disprove it?