An inequality involving powers of sums of powers

A Superelliptic Equation Involving Alternating Sums of Powers

In this short note, we solve completely the Diophantine equation 1 − 3 + 5 − · · · + (4x− 3) − (4x− 1) = −y, for 3 ≤ k ≤ 6. This may be viewed as a “character-twisted” analogue of a classic equation of Schaffer (in which context, it was previously considered by Dilcher). In our proof, we appeal primarily to techniques based upon the modularity of Galois representations and, in particular, to a ...

A Sharp Double Inequality for Sums of 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...

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...