While the approximation,
e^(pi*sqrt(163)) ~ 640320^3 + 743.99999999999925...
is quite well-known, not so well-known is this one,
e^(pi*sqrt(163)) ~ (x^3-6x^2+4x-2)^24 - 24.000000000000001051...
or the real root of the cubic polynomial equated to zero and raised to the 24th power is almost Ramanujan's Constant. Interestingly enough, it also misses it by an amount exceedingly close to 24.
This is not an isolated case, as will be shown by finding other polynomials with relatively small coefficients and similar results. One such polynomial, a nonic, seems to be the keenness known so far...
