Collatz Stopping Times

Math 242 practice problems

Name: _________________

Learning Goals: Use computation to explore mathematical patterns and test conjectures.

Your task: Complete the two exercises from the text, reprinted below.

Due: Wednesday, March 13, at 5:00pm. Upload your solution as a Mathematica notebook to the Collatz Stopping Times assignment (Moodle link).

Grading: Problem 1 requires you to find 17 consecutive positive integers with the same stopping time to earn a Satisfactory grade. For Problem 2, a solution will be considered Satisfactory if you provide either evidence in support of the stated conjecture or a counterexample that disproves the conjecture.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, email Prof. Wright, or visit the help session.

Problem 1: Exercise 3.26

Find 17 consecutive integers with the same stopping time. That is, find integers n,n+1,n+2,...,n+16 such that stop(n)=stop(n+1)=stop(n+2)=⋯=stop(n+16).

Problem 2: Exercise 3.30

Let α(n) be the number of odd integers in the Collatz trajectory of n, not including 1. For example, α(3)=2 and α(7)=5. Rafael Ruggiero conjectured that the stopping time satisfies stop(n)=⌈(Subscript[log, 2])(n·(6^(α(n))))⌉. Investigate Ruggiero’s conjecture. Either produce computational evidence in support of the conjecture, or find a counterexample. Note that ⌈x⌉ denotes the “ceiling” of x, which is the least integer greater than or equal to x.