The Syracuse Problem

For a given nonzero natural integer n=n0, one constructs a list of integers n1, n2, n3,... in the following way. If n0 is even then n1=n0/2. If n0 is odd then n1=3n0+1. The next element, n2, of the list of integers is determined by the same rule applied to n1: if n1 is even then n2=n1/2. If n1 is odd then n2=3n1+1.

For example, if n0=1 then n1=4, n2=2, n3=1, n44, etc. We have a cycle: 1,4,2,1,4,2,1,4,2,1,...

Let us start with another value for n0. If n0=3, we find the list 10,5,16,8,4,2,1,4,2,1,4,2,1,...

Let us try yet another one, starting with 7: 22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,...

It seems that we eventually fall into the cycle 1,4,2,1,... This is the Syracuse Problem, do we always end up in the cycle 1,4,2,1,.. whatever initial value we take? In fact, often the problem is posed by asking whether we arrive eventually at 1 starting with any nonzero natural integer.

It is interesting to write the integers of the list in binary representation. Graphically, a black disc will be a 1 and a white disc will be a 0 digit. Drawing the numbers in a spiral, least significant bits in the interior, most significant bits at the rim, we get the following kind of dynamics:

The starting value here is 18446744073709551617 which happens to be equal to 226+1. This explains why at the start we only have two black discs. the movie stops when we reach 1. If one wants to control the movie, one can install gif reader and activate it on this page.

Here is another one.

Last modification: Décembre 23, 2014