An Analysis of the Collatz Conjecture Alexander Townsend

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 The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937 at Syracuse University. The conjecture is also known as the 3 n + 1 conjecture, Ulam Problem , Kakutani's problem, Thwaites conjecture , Hasse's algorithm, or the Syracuse problem; In the following example, we create an infinite flowchart to show each member of any sequence behaves under the modified Syracuse function. The infinite sequence helps us get a better understanding of what equivalence classes the odd number belong to and prepare us for our next result. We consider the following proposition. DEFINITION In order to better understand what classes of integers further satisfy the Collatz condition we look at each sequence of integers that converge to 1 under the modified Syracuse function through a Python code. 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 considerably more well behaved, enough to categorize them into certain equivalence classes. In order to better understand the sequences, we also graph the 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 4 k-1, 8 k+1 and … the following function gives all odd numbered terms in the Collatz conjecture References: i. Applegate, D. , Largarias, J. , The 3 x+1 semigroup, Journal of Number Theory, 117, (2006), 146 -159. ii. Lagarias, J. , The 3 x+1 Problem: An Annotated Bibliography (2000 -2009) Preprint: arxiv: math. NT/0608208. Aug. 27, 2009, v 5. 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.