Representation by sums of unlike powers

Sums of Powers by Matrix Methods

This equation may be viewed as reducing the sum of n terms of the power sequence to a linear combination of the first r + 1 terms. Accordingly, there is an implicit assumption that n > v. Note that the matrix appearing as the middle factor on the right side of this equation is lower triangular. The zeros that should appear above the main diagonal have been omitted. The nonzero entries constitut...

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.

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...