On sums of powers

Proceedings - Mathematical Sciences -     

Equal sums of sixth powers

The problem of solving x⁶ + y⁶ + z⁶ = u⁶ + v⁶ + w⁶, subject to certain auxiliary conditions, is reduced to a single cubic. In this way some earlier known solutions are inserted into an infinite cyclic group lying on the cubic.     

surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .

A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . The main result follows by a deformation argument.

     

On sums which are powers

The cardinality of sets A is estimated under the conditions that every element of the sum set A+A is a power resp. powerful number (n is said to be powerful if p n implies p ² n). Subset sums with these properties are also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

On sums of powers of integers

K. Thanigasalam has shown that for any positive integer k the sequence of positive integers represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^5\text{+ x}{}_1{}^{\mathrm{k}}$ has positive density. Here we prove that the asymptotic density of this sequence is 1, a similar result is proved for the sequence represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^6\text{+ x}{}_1{}^{\mathrm{k}}$.     

Compositions of sums of absolute powers

We obtain an explicit formula for then-dimensional volumes of certain bodies, calledoddballs hereinafter. An oddball is a bodyG = {x εR ⁿ :f(x) ≤ 1}, wheref:R ⁿ →R is anoddball function. Oddball functions are defined by way of the following construction: We begin with the class of functionsf of the formf(x ₁, ...,x _k ) = |x ₁|^α + |x ₂|^β + ... + |x _k|^γ. Herek may be any positive integer, and is not fixed. The Greek exponents are arbitrary positive real numbers. We extend this class by permitting any finite number of substitutions among functions in the class. Finally, we extend the substitution-enlarged class by permitting linear formsy _i = Σ _j b _ij x _j to replacex _i 's, the transformations being nonsingular. Thus, if det(b _ij ) ≠ 0, the oddball function $$f(x_1 ,x_2 ,x_3 ,x_4 ,x_5 ,x_6 ) = ((|y_1 |^\alpha + |y_2 |^\beta )^\tau + (|y_3 |^\gamma + |y_4 |^\phi + |y_5 |^\psi )^\delta )^\mu + |y_6 |^\eta $$ is a fairly typical example. We also consider the number of lattice points in certain types of oddballs, as well as their latticepacking densities. Neither do oddballs include thesuperballs discussed elsewhere by this and other authors, nor is every oddball a superball.     

Relations among sums of reciprocal powers

Some formulas relating different classical sums of reciprocal powers are derived. These relations can be written in a very compact way by means of certain numbers which include Catalan’s constant and satisfy simple summation formulas. This paper has been partially supported by a DGESIC grant PB97-0342.     

Equal sums of four seventh powers

In this paper, the method used to find the smallest, nontrivial, positive integer solution of is discussed. The solution is

Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.

     

Bernoulli numbers and sums of powers

There have been many studies of Bernoulli numbers since Jakob Bernoulli first used the numbers to compute sums of powers, 1 ^p + 2 ^p + 3 ^p + ··· + n^p , where n is any natural number and p is any non-negative integer. By examining patterns of these sums for the first few powers and the relation between their coefficients and Bernoulli numbers, the author hypothesizes and proves a new recursive algorithm for computing Bernoulli numbers, sums of powers, as well as m-ford sums of powers, which enrich the existing literatures of Bernoulli numbers.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

Generalized Cochrane sums and Cochrane-Hardy sums

In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.