A new algorithm to generate a formula for the sum of integer powers

Abstract

A new algorithm is proposed to generate a formula for the sum of integer powers Sp(n) = Σnk=1kp for an arbitrary positive integer p. This formula plays an important role in scientific computation, numerical analysis, complexity analysis, academic research, and even in teaching calculus. Faulhaber's formula indicates that Sp(n) can be expressed as a (p + 1)th degree polynomial. A special linear system is constructed and then solved to fit this polynomial through the Gaussian Elimination method. This study shows that this new algorithm is more efficient, having a polynomial time O(p3) complexity. In the implementation procedure, this algorithm does not use any complicated Bernoulli numbers, Stirling numbers, integrals, differentiations, or recursion methods. Maple software code is used to illustrate how the new algorithm works, and the coding of this algorithm has only five lines. For a power of 20 or 100, the computer execution CPU (Intel Processor running @ 3GHz) time takes only 0.062 seconds or 16.396 seconds respectively.