Proving theorems and special cases (Part 8): The Collatz conjecture

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve already seen that a conjecture could be true for the first 40 cases or even the first $10^{316}$ cases yet ultimately prove false for all cases.

For the next few posts, I thought I’d share a few of the most famous unsolved problems in mathematics… and just how much computational work has been done to check for a counterexample.

3. The Collatz conjecture (see here and here for more information) is an easily stated unsolved problem that can be understood by most fourth and fifth graders. Restated from a previous post:

Here’s the statement of the problem.

• If the integer is even, divide it by $2$. If it’s odd, multiply it by $3$ and then add $1$.
• Repeat until (and if) you reach $1$.
Here’s the question: Does this sequence eventually reach $1$ no matter the starting value? Or is there a number out there that you could use as a starting value that has a sequence that never reaches $1$?
For every integer less than $5 \times 2^{60} = 5,764,607,523,034,234,880$, this sequence returns to 1. Of course, this is not a proof that the conjecture will hold for every integer.