• Summation of Numerical Powers. The discovery of closed formulas for discrete sums of numerical powers, motivated by application to approximations for solving area and volume problems in calculus, is probably the most extensive thread in the development of discrete mathematics, spanning the period from ancient Pythagorean interest in patterns of dots to the work of Euler on a general formula for discrete summations. Initial sources from classical Greek, Indian, and Arabic traditions include Archimedes’ [1] determination of the closed formula for a sum of squares, then Nichomachus, Aryabhata, and al-Karaji on sums of cubes, and al-Haytham on the sum of fourth powers. In the seventeenth century Fermat (1601–1665) [23] claimed that he could use the “figurate numbers” to solve this, “perhaps the most beautiful problem in all of arithmetic”. Fermat’s work was followed shortly by Pascal’s [54] extensive treatise on this topic, which produced the first explicit recursive formula for sums of powers in any arithmetic progression using binomial coefficients, and which can be proved by mathematical induction. Pascal writes “I will teach how to calculate not only the sum of squares and of cubes, but also the sum of the fourth powers and those of higher powers up to infinity”. This material will easily make several one-week projects at various levels, and is intended for courses in discrete mathematics and combinatorics, with PI David Pengelley as the primary author.
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