The great mathematician Paul Erdös (1913–1996) described the attempt to find a solution to the Collatz problem as »absolutely hopeless«. The problem sounds so simple in its basic assumptions that even elementary school children understand it. It begins with a sequence that you build up according to these rules: Take a number; if it is even, divide it by two; if it’s odd, multiply it by three and add one. You repeat that over and over again. For example, you can start with 19 and get: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, … Or with twelve: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, … As soon as the sequence ends up with the one, it becomes periodic, i.e. it repeats itself, because according to the arithmetic rule follows: 1, 4, 2, 1, 4, 2, 1 and so on.
The resulting Collatz problem, also known as the Collatz conjecture, is: Every natural number that you start with will inevitably end up with one at some point. Accordingly, every sequence of numbers would have a periodic end. In recent decades, a number of experts and people with an affinity for mathematics have tried to solve what is supposedly the simplest problem in the subject – but in vain. In this column, however, I will not devote myself to the failed proof ideas, but will show that the number pi also appears in the Collatz conjecture!
Syracuse or Syracuse?
Pi appears in the strangest of environments, such as billiards, fractals, the game of life, and infinite sums. And indeed, the number of circles can also be found in the Collatz problem. Sometimes referred to as the Syracuse Conjecture, it is now suspected that there may be a connection to Archimedes of Syracuse. Because Pi is also called Archimedes’ constant, because Archimedes was the first to design an algorithm to calculate the digits of Pi. But Syracuse is not the link we are looking for between pi and the Collatz conjecture: while “Syracuse” in the case of Archimedes refers to his birthplace in Sicily, “Syracuse” in the name of the mathematical problem has a completely different origin, which with its becoming known has to do.