Given a natural number n, the Collatz sequence it generates is the following: if n is even, then divide it by 2, if n is odd, then multiply it by 3 and add 1; and now iterate this procedure. The Collatz conjecture states that you will always end up with the number 1 after finitely many steps.

There is an interesting article about the Collatz conjecture at the QuantaMagazine that I wanted to share with you (since I enjoyed reading it): Mathematician Terence Tao and the Collatz Conjecture.

It is based on recent results of Terence Tao about the conjecture (arXiv:1909.03562). He proved the following: if \(c_{\mathrm{min}}(n)\) denotes the minimal number occurring in the Collatz sequence that starts from \(n\), then for any function \(f\) on the natural numbers with \(\lim_{n \to \infty} f(n) = +\infty\) we have for almost all \(n \in \mathbb{N}\) that \(c_{\mathrm{min}}(n) \le f(n)\). For example, we have \(c_{\mathrm{min}}(n) \le \log \log \log \log n\) for almost all \(n\). This is since many years the best progress on the conjecture that we had.