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  • $\begingroup$ I agree. Maple tells me R^10 is not that close to an integer. Try smaller powers. $\endgroup$
    – Michael Lugo
    Commented Nov 10, 2009 at 2:21
  • $\begingroup$ Yes - I tried everything up to 12. R^n seems to drift away from being an integer. $\endgroup$
    – Sam Nead
    Commented Nov 10, 2009 at 2:26
  • $\begingroup$ Ah, perhaps I see - even for R^2 to be close to an integer is surprising. $\endgroup$
    – Sam Nead
    Commented Nov 10, 2009 at 2:32
  • $\begingroup$ Right. Because $R$ differs from an integer by much less than $1/R$, it's already surprising that $R^2$ is close to an integer. The reason for this is that $N\epsilon$ is also close to an integer. But this logic doesn't seem to generalise well enough to get to $R^5$. $\endgroup$
    – Kevin Buzzard
    Commented Nov 10, 2009 at 7:43