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The use of the axiomatic method, by showing clearly the source of each proposition and by showing which were the essential hypotheses and the superfluous hypotheses, has revealed unsuspected analogies and permitted extended generalizations; the origin of the modern developments of algebra, topology and group theory is to be found only in the employment of axiomatic methods.

Jeremy Gray, The Hilbert Challenge (2000)

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Sums of Powers of Positive Integers

Introduction

Students often encounter formulas for sums of powers of the first n positive integers as examples of statements that can be proved using the Principle of Mathematical Induction and, perhaps less often nowadays, in Riemann sums during an introduction to definite integration.  In either situation, they usually see only the first three such sum formulas,

$$1 + 2 + 3 + \cdots + n = {{n(n + 1)}\over 2}$$

$$1^2 + 2^2 + 3^2 + \cdots + n^2 = {{n(n + 1)(2n + 1)} \over 6}$$

and

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = {{n^2 (n + 1)^2 } \over 4}$$

for any positive integer n.

Formulas for sums of integer powers were first given in generalizable form in the West by Thomas Harriot (c. 1560-1621) of England.  At about the same time, Johann Faulhaber (1580-1635) of Germany gave formulas for these sums up to the 17th power, far higher than anyone before him, but he did not make clear how to generalize them.  Pierre de Fermat (1601-1665) often is credited with the discovery of formulas for sums of integer powers, but his fellow French mathematician Blaise Pascal (1623-1662) gave the formulas much more explicitly.  The Swiss mathematician Jakob Bernoulli (1654-1705) is perhaps best and most deservedly known for presenting formulas for sums of integer powers to the European mathematical community.  His was the most useful and generalizable formulation to date because he gave by far the most explicit and succinct instructions for finding the coefficients of the formulas.

In this article, we first recount some of the early history of formulas for sums of integer powers.  We then explore how each of the mathematicians listed above developed, understood, and presented their formulas.  Along the way, we shall see that the phrase “general formula” is relative terminology:  during the late sixteenth and early seventeenth centuries mathematicians were just beginning to replace verbal descriptions by symbolic representations and we may be surprised to discover that some mathematicians’ “formulas” were given entirely in words and/or were given just for the first few exponents with the remaining cases covered by “et cetera”.  At the end of each section, we present exercises and activities designed to help students develop a deeper understanding of the ideas and methods of each of the mathematicians we discuss.

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Beery, Janet, "Sums of Powers of Positive Integers," Loci (February 2009), DOI: 10.4169/loci003284

Karaji's solution

I just wonder if the Karaji Generalized his method of solutions? Is there any information about that. For instance did he found sum of fifth powers?

Al-Karaji and higher powers by Janet Beery (posted: 01/20/2010 )
My understanding is that there is no evidence that al-Karaji derived a "formula" for the sum of the fourth, fifth, or higher powers. His justification for his formula for the sum of the cubes was ingenious but al-Haytham's idea seems more readily generalizable.

On the symmetry of the resulting polynomials

== Conjecture == It would be interesting to know that * sums of odd powers can be factored as a polynomial on (n Ã¢â€¹â€¦ (n + 1) / 2) and is therefore symmetric with the center at (Ã¢Ë†â€™Ã‚Â½); and that * sums of even powers can be factored to a function like above multiplied by (2 Ã¢â€¹â€¦ n + 1) and therefore antisymmetric with the center at (Ã¢Ë†â€™Ã‚Â½). It would be interesting to know whether any rule concerning the polynomials thus obtained.

Replies:

Re: On the symmetry of the resulting polynomials by Rick Mabry (posted: 12/30/2011 )
Christopher, see the following article, especially section 3 (Faulhaber Polynomials). Although my browsers don't let me view the code you pasted, I think the article confirms some of what you're getting at (some of which might be gleaned from page 8 of the article we're viewing). A. F. Beardon, Sums of Powers of Integers, American Mathematical Monthly, Vol. 103, No. 3 (Mar., 1996), pp. 201-213.

More about symmetry: by Janet Beery (posted: 01/03/2013 )
The most succinct, beautiful, and perhaps even historically plausible explanation I've seen for the symmetry of the Faulhaber polynomials about -1/2 was in a recent article by Reuben Hersh in the College Math Journal 43:4 (Sept. 2012), pp. 322-4. His approach? Extend the polynomials to negative values of n.