I seem to have come across a curious little question whilst wondering if a generalised quadrangle can have a product structure, and it comes down to a question about numbers. I’ve been unable to figure it out, but I’m sure it’s not hard. Can somebody prove the following?
Let be an integer greater than 2 and suppose is a nontrivial power of a positive integer. Then .
I might put the problem on Mathoverflow. Peter Mac had a good crack at it; thanks Peter!
It’s been answered beautifully on MathOverflow:
http://mathoverflow.net/questions/58697/a-geometric-series-equalling-a-power-of-an-integer
and the solution goes back to work of Nagell and Ljunggren. For a good reference on this problem, see the recent paper of Bugeaud and Mihailescu:
Click to access PredaYann1.pdf