Problem of the Week: Powerful Numbers

Problem of the week

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 panel of faculty judges, 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.


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! Send solutions to or put solutions in the marked envelope in the hallway outside Glatfelter 215. This problem was posted on Friday, March 22 and solutions are due on Friday, March 29 by 5:00 p.m.

THE PROBLEM: Powerful numbers
Let m be an integer (that is, a whole number — perhaps positive, perhaps negative, perhaps zero). Let us say that m is a SPA number if there are positive whole numbers k and j such that

m = (m^k + m^j)/2.

QUESTION. Find all the SPA numbers. In your solution, explain how you know your list is complete.

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Author: Gettysburgian Staff

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