What I'm actually trying to do is efficiently calculate Faulhaber(N, E) modulo M (or at least Faulhaber(N, E) - the modulo operation could be worked in later), where N, E, and M can be arbitrarily large. From what I have read, it seems that this is generally expressed in terms of either a polynomial to the degree of N+1 (memory-expensive) or some recursive relationship (time-expensive). Is there any way that this can be done that requires neither very much memory nor time?
Code:#include <cmath> #include <complex> bool euler_flip(bool value) { return std::pow ( std::complex<float>(std::exp(1.0)), std::complex<float>(0, 1) * std::complex<float>(std::atan(1.0) *(1 << (value + 2))) ).real() < 0; }
Since the exponent changes, you'll be forced to use something like this:
Faulhaber's Formula -- from Wolfram MathWorld
(the first equation with the Kronecker delta etc, not the individual cases)
Power Sum -- from Wolfram MathWorld
may also be helpful.
