2nd ed. January 21, 2018; 1st ed. Feb 8th, 2010
Abstract
This three part paper explores solving the sum of powers problem using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers .
Part 1 (this paper) solves the problem using recurrence relations in a way which a high school student could emulate for small . In Part 2, we develop a general recursive solution that works for arbitrary , from which we can build a table of values to assist in finding the coefficients of the solution polynomial, coefficients that are precisely the Bernoulli numbers discovered in 1713. In Part 3, we show how by transforming the problem into a linear system, we may obtain a direct (non-recursive) solution which directly calculates the Bernoulli number for any power . Source code is provided for all solutions.
Readers who are interested in this topic are referred also to lovely paper by Bearden (March 1996, American Mathematical Monthly), which tells the mathematical story and fills in the history (thanks to a reader for this great reference).
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