Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli

Abstract

In 1636 Pierre de Fermat wrote to Gilles Personne de Roberval that he could solve "what is perhaps the most beautiful problem of all arithmetic." By this he meant the two millennium old problem of finding a closed formula for a sum of k-th powers of integers from 1 to n, which stretches from the Pythagoreans and Archimedes through many mathematicians, including Nicomachus, Āryabhata, al-Karajī, ibn al Haytham, Fermat, Pascal, Bernoulli, and Euler, culminating in the discovery of the Bernoulli numbers and the Euler-MacLaurin summation formula, which enabled Euler to rise to fame by first guessing and then proving that the sum of the reciprocal squares is (π^2)/6. Here we introduce and provide for instructors a student project based on original sources in this story by Fermat, Pascal, and Bernoulli, designed for a course in combinatorics or advanced discrete mathematics.