Sums of powers and majorization

Sums of powers via integration

Sum of powers 1 + · · · + n, with n, p ∈ N and n ≥ 1, can be expressed as a polynomial function of n of degree p + 1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing coefficients of Faulhaber formulae is presented. The correctness of the algorithm is proved by giving a recurrence relation on Faulhaber formulae.

Sums of Squares, Cubes, and Higher Powers

Exponential Sums modulo Prime Powers

where p is a prime power with m ≥ 2, χ is a multiplicative character (mod p), epm(·) is the additive character, epm(x) = e 2πix pm , and f, g are rational functions with integer coefficients. It is understood, that the sum is only over values of x for which g and f and both defined as functions on Z/(p), and g is nonzero (mod p). The sum is trivial if f and g are both constants, so we shall alw...

On sums which are powers

On powers that are sums of consecutive like powers

1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example...