An Analysis of the Collatz Conjecture

Alexander Townsend; William Venables| Dr. Feroz Siddique

The Collatz Conjecture is an unsolved problem in mathematics and is named after Lothar Collatz who is believed to have proposed it in one of his papers in 1937. This problem is also called the Hailstone problem because the rising and falling numbers symbolically represents the motion of a hailstone in a storm rising due to updrafts and falling due to gravity. We highlight what the problem is, highlight well known results as and state some of the approaches we have taken to delineate interesting properties enjoyed by each sequence of integers converging to 1 through the Collatz function. According to Paul Erdős , “Mathematics is not yet ready for such problems.”

INTRODUCTION In the following example, we create an infinite flowchart to show how each member of any sequence behaves The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz, who first under the modified Syracuse function. The infinite sequence helps us get a better understanding of what proposed it in 1937 at Syracuse University. The conjecture is also known as the 3 n + 1 conjecture, Ulam equivalence classes the odd number belong to and prepare us for our next result. Problem , Kakutani's problem, Thwaites conjecture , Hasse's algorithm, or the Syracuse problem;

DEFINITION We consider the following proposition .

Proposition: For each integer n, the sequence of integers obtained by repeated application of the modified Syracuse function must always be of the form 6 1. Proposition: Every integer of the form reduces to 1 after finitely many applications of the Collatz function.

The conjecture can be checked for all values up to 20 ∗ 2 (Oliviera De Silva, 2008). The conjecture can In order to better understand what classes of integers further be proved to be false if we obtain a sequence of numbers, starting with the initial integer, that does not satisfy the Collatz condition we look at each sequence of integers end with 1. It is possible that such a sequence enters a repeating cycle that does not include 1. that converge to 1 under the modified Syracuse function through a Python code. Stefan Andrei and Christian Masalagiu proved that the Collatz Conjecture is true for the following class of integers:

We graph the stopping time for all integers up to 10000. Observe the small continuous dashes that indicates how a set of integers in certain neighborhood have the same set of integers in its sequence under the modified Syracuse function.

Maximum value in each sequence converging to 1 in Collatz function. We consider the following modified Collatz function which only gives us odd integers at each step of the Collatz sequence. We feel that if we remove the even integers from each step, then the sequences should be In order to better understand the sequences, we also graph the considerably more well behaved, enough to categorize them into certain equivalence classes. highest power of 2 in the sequences for each integer. Here we show up to the first 20000. The pattern is very similar for higher values of x. MODIFIED SYRACUSE FUNCTION DEFINITION

Classifying all odd numbers in the form 4k-1, 8k+1 and … the following function gives all odd References: numbered terms in the Collatz conjecture i. Applegate, D., Largarias, J., The 3x+1 semigroup, Journal of Number Theory, 117, (2006), 146-159. ii. Lagarias, J., The 3x+1 Problem: An Annotated Bibliography (2000-2009) Preprint: arxiv:math.NT/0608208. Aug. 27, 2009, v5. iii. Andrei, S., Masalagiu A., About the Collatz conjecture, Acta Informatica, 35, (1998), 167-179. iv. Couture,O., Unsolved Problems in Mathematics, Whiteword publications, (2012). v. Motta,F., Oliviera,H., Catalan,C., An Analysis of the Collatz Conjecture.

We thank the Office of Research and Sponsored Programs for supporting this research, and Learning & Technology Services for printing this poster.