RECURSIVE DETERMINATION OF THE ENUMERATOR FOR SUMS OF THREE SQUARES

For each nonnegative integer n, r3(n) denotes the number of representations of n by sums of three squares. Here presented is a two-step recursive scheme for computing r3(n), n≥ 0.

The outstanding result about r 3 (n), n ∈ N, was first presented by Legendre.We state his result in the following theorem.
Based on his theory of ternary quadratic forms, Gauss gave the first complete proof of this theorem in his now famous book Disquistiones Arithmaticae.
In this note, our major objective is to give a two-step recursive determination of the sequence r 3 (n), n ∈ N.This is accomplished by the following two results.
Theorem 1.4.For each n ∈ N, (1.4) As before, summation extends as far as the arguments of q 0 remain nonnegative.

Proof of Theorem 1.4.
Our proof is based on the following two identities, each of which is valid for each complex number x such that |x| < 1.
Now, we expand the product of the two series, and subsequently equate coefficients of like powers of x to prove our theorem.
Our recursive two-step algorithm proceeds as follows: (i) use the recursive determination of q 0 in Theorem 1.3 to compile a table of values of q 0 , as in Table 2.1, (ii) in terms of these computed values of q 0 we then utilize Theorem 1.4 to compile a table of values of r 3 , as in Table 2.2.Concluding remarks.The brief Tables 2.1 and 2.2 are compiled to show the effectiveness of our procedure.In terms of machine computation we observe that for a fixed but arbitrary choice of n ∈ P, each of Theorems 1.3 and 1.4 requires 0(n 3/2 ) running time.Legendre's Theorem 1.2 would provide an excellent check on the accuracy of computation.
For given n ∈ P, there are formulas which express r 3 (n) in terms of certain divisor functions; and, also in terms of Jacobi symbols.However, factorization of arguments of the divisor functions and denominators of Jacobi symbols is required before these expressions can be utilized.By comparison our procedure is entirely additive in character.In a word, no factorization is required.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: