Hilbert’s 10.5th Problem

via rjlipton.wordpress.com

Hilbert’s Tenth Problem (HTP) is the question raised by David Hilbert at his famous 1900 Paris lecture on whether or not there is an algorithm to decide whether a Diophantine equation has a solution over the integers:

Java Code examples - Bernoulli polynomial function

 /**
   * return nth Bernoulli polynomial of x using Fourier expansion
   * ref: Abramowitz & Stegun 23.1.16 pp 805
   * @param n
   * @param x
   * @return
   */
  public static double BernoulliP (int n, double x)
  {
    // for n>1, 1>=x>=0
    // for n=1, 1>x>0
    
    // for x=0 return the nth Bernoulli number
    if (x==0.0) {
      return BernoulliB(n);
    }
        
    double K = -2*(factorial(n))/StrictMath.pow((2*StrictMath.PI), n);
    
    double S = 0.0;
    for (int k=1; k<100001; k++)
    {
      S += 
        StrictMath.cos((2*StrictMath.PI*k*x)-(.5*StrictMath.PI*n))/
          StrictMath.pow(k, n);
    }
    
    return K*S;
  }