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    $\begingroup$ I think that what you're saying FC is that rather than just raising the q-expansion of j to the 5th power and then noting that the bounds aren't good enough and giving up, we should be writing down some more careful 5th degree poly in j, which would be good enough. And the fact that you didn't do this is some sort of indication that the correct poly is a little tricky to find? $\endgroup$
    – Kevin Buzzard
    Commented Nov 10, 2009 at 8:58
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    $\begingroup$ So pari can do LLL. Using it I get $126476476039j-1503930055j^2+3945885j^3-3519j^4+j^5=q^{-5} + 201q^{-4} - 6879q^{-3} - 2463q^{-2} - 6323q^{-1} - 13080 - 7358q + 347786775737915540q^2 + O(q^3)$ where the coefficients in $q^n$, $n\geq3$ aren't growing as fast as $10^{17n}$. This shows that if $R^n$ is close to an integer for $n=1,2,3,4$ then it's also close for $n=5$. $\endgroup$
    – Kevin Buzzard
    Commented Nov 10, 2009 at 9:20