Fun Facts about Collatz 3x+1 Sequences

This mathematical essay explores the hidden regularities of Collatz sequences, which at first sight seem rather chaotic. Their behavior is however determined by a simple threshold number and by binary signatures, which capture the percentage of down versus up movements. Their starting elements are described by "primal numbers", which share properties with prime numbers. The relationship with prime numbers and some famous theorems and unsolved problems are explored, such as Bertrand's postulate and de Polignac's second conjecture. The twin prime conjecture is rephrased as an example of the pigeonhole principle. Graphically, a Collatz sequence will be interpreted as a walk on a fractal, the Cantor set.