For certain choices of the coefficients a, b, c the solutions of the Diophantine equation ax+by = cz in Gaussian integers satisfy xy = 0.

We present a new criterion for indecomposability of polynomials over Z. Using the criterion we obtain general finiteness result on polynomial Diophantine equation f(x) = g(y).

Reduced-order observer design has a long history spanning decades and involving various researchers in the control systems society. The first results on this problem were presented by Luenberger [1]. Thereafter, several papers have been presented examining the problems from different perspectives. One of these approaches is the Sylvester equation approach. This technique aeals mainly with the c...

In 1997, Darmon and Merel proved the stunning result that the Diophantine equation x + y = z has no nontrivial integer solutions for n ≥ 4. This can be interpreted as saying that if {vn} represents a Lucas sequence of the second kind, defined by a quadratic polynomial with rational roots, then the equation vn = x , with x an integer, implies that n ≤ 3. The goal of the present paper is to prove...

We give a combinatorial proof for a second order recurrence for the polynomials pn(x), where pn(k) counts the number of integer-coordinate lattice points x = (x1, . . . , xn) with ‖x‖ = ∑n i=1|xi| ≤ k. This is the main step to get finiteness results on the number of solutions of the diophantine equation pn(x) = pm(y) if n and m have different parity. The combinatorial approach also allows to ex...

A study of the diophantine equation v2 = 2u4 − 1 led the authors to consider elliptic curves specifically over Q(i) and to examine the parallels and differences with the classical theory over Q. In this paper we present some extensions of the classical theory along with some examples illustrating the results. The well-known diophantine equation v2 = 2u4 − 1 , has, ignoring signs, only two integ...

A difficulty in solving linear Diophantine systems is the rapid growth of intermediate results. Rosser’s algorithm for solving a single linear Diophatine equation is an efficient algorithm that effectively controls the growth of intermediate results. Here, we propose an approach to generalize Rosser’s algorithm and present two algorithms for solving systems of linear Diophantine equations. Then...