# Problem of the Week: Spring 2024, Number 3

Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s).  Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button.  The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.

Solutions Due: Saturday, February 10, 5:00 p.m.

Send solutions to bkennedy@gettysburg.edu.

The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Weekly winners will receive a POW victory button, and the best-performing students of the semester will receive the Paul Mugabi problem-solving award.

This is the greatest prime puzzle!

Start with any two positive whole numbers a and b. Then make a sequence a, b, c, d, . . . in the following way:

• c is the largest prime number dividing (a + b);
• d is the largest prime number dividing (b + c);

. . . and so on. Example: if we begin with 5 and 6, the resulting sequence starts like this: 5, 6, 11, 17, 7, 3, 5, 2, . . .

Describe, as completely as you can, all possibilities for how such a sequence can look “in the long run.”

Author: Gettysburgian Staff