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  • $\begingroup$ I am by no means an expert on this material. I am just easily amused by tricks that people can play with power series, and enough so to want to see an answer to this question. $\endgroup$
    – Michael Lugo
    Commented Nov 9, 2009 at 21:25
  • $\begingroup$ Writing $R=N-\epsilon$ you can see easily that $R^2$ is not close to $N^2$, so that's why $R^5$ is not close to $N^5$! $\endgroup$
    – Kevin Buzzard
    Commented Nov 9, 2009 at 21:52
  • $\begingroup$ Of course. I only specifically singled out the fifth-power case because that's the case that the original question was about. $\endgroup$
    – Michael Lugo
    Commented Nov 9, 2009 at 22:00
  • $\begingroup$ Typo: q=-exp(-pi.sqrt(163)) in the above (missing minus sign). Fix the typo and then I delete the comment, and no-one ever knew. $\endgroup$
    – Kevin Buzzard
    Commented Nov 9, 2009 at 22:01
  • 1
    $\begingroup$ Ben has just written up a lot of the details of the standard proof that the constant is almost an integer. I think one needs to go a little deeper into the theory to answer the question at hand. $\endgroup$
    – Kevin Buzzard
    Commented Nov 9, 2009 at 23:42