IBM Ponder This #084 · April 2005

Puzzle for April 2005

This month's puzzle is sent in by Max Alekseyev. It appeared in a
Russian mathematics olympiad for university students.
It was originally due to Marius Cavache.

We are not asking for submission of answers this month.

Problem:

"b^k" denotes b raised to the k power.

Let a,b be integers greater than 1. Clearly if b=a^k for some positive
integer k, then ((a^n)-1) divides ((b^n)-1) for each positive integer n.
Prove the converse: If ((a^n)-1) divides ((b^n)-1) for each positive
integer n, then b=a^k for some positive integer k.

Solution

Best opened after a real attempt

To be added.