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: Friday, March 18, at 5:00pm. Upload your solution as a Mathematica notebook to the Collatz Stopping Times assignment (Moodle link).

Grading: If you complete each problem below, you will receive a Satisfactory grade.

Need help? Review the notes from class, which are posed on the course web site. If you still need help, email Prof. Wright.

1. Exercise 3.30: 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).

2. Exercise 3.34: 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.