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An exercise in number theory?

March 16, 2011

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.

 

 

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3 Comments leave one →
  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

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