On Hirzebruch sums and a theorem of Schinzel

on direct sums of baer modules

the notion of baer modules was defined recently

Faulhaber's theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b, a+2b, . . . , a+nb is a polynomial in na+ n(n + 1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, b...

On Coverings of the Integers Associated with an Irreducibility Theorem of A. Schinzel

Pfister’s Theorem on Sums of Squares

A similar 4-square identity was discovered by Euler in 1748: (x1 + x 2 2 + x 2 3 + x 2 4)(y 2 1 + y 2 2 + y 2 3 + y 2 4) = (x1y1 − x2y2 − x3y3 − x4y4) + (x1y2 + x2y1 + x3y4 − x4y3) + (x1y3 − x2y4 + x3y1 + x4y2) + (x1y4 + x2y3 − x3y2 + x4y1). This was rediscovered by Hamilton (1843) in his work on quaternions. Soon thereafter, Graves (1843) and Cayley (1845) independently found an 8-square ident...

The Hurwitz Theorem on Sums of Squares

This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...