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 $s$ be an integer greater than 2 and suppose $(s+1)(s^2+1)$ is a nontrivial power of a positive integer. Then $s = 7$.

1. March 17, 2011 8:24 am

I might put the problem on Mathoverflow. Peter Mac had a good crack at it; thanks Peter!

2. March 18, 2011 8:48 am

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:

http://www-irma.u-strasbg.fr/~bugeaud/travaux/PredaYann1.pdf