Let 6 denote the positive root of the equation xs + x2 — 2x — 1 = 0; that is, 8 = 2 cos(27r/7). The main result of the paper is the evaluation of the constant lim supm-co min M2\x + By + 02z|, where the min is taken over all integers x, y, z satisfying 1 g max (\y\, |z|) g M. Its value is (29 + 3),/7 = .78485. The same method can be applied to other constants of the same type.

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. 1991 Mathematics Subject Classification: 11D09.

Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <...

Determining whether a Diophantine equation has a solution or not is the most important challenge in solving this type of problems. In this paper a special computational device which uses light rays is proposed to answer this question, namely check the existence of nonnegative solutions for linear Diophantine equations. The way of representation for this device is similar to an directed graph, h...

Let a1, a2, . . . , ak be positive and pairwise coprime integers with product P . For each i, 1 ≤ i ≤ k, set Ai = P/ai. We find closed form expressions for the functions g(A1, A2, . . . , Ak) and n(A1, A2, . . . , Ak) that denote the largest (respectively, the number of) N such that the equation A1x1 + A2x2 + · · · + Akxk = N has no solution in nonnegative integers xi. This is a special case of...

In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is a(i, j) = 0, 1, . . . , M − 1, with M being a prime number, we reduce the equations modulo M ...

What can be said about the set E ~ of solutions in nonnegative integers to a system of linear equations with integer coefficients? For many purposes, such as those of linear programming, this question has been adequately answered. However, when this question is regarded from the vantage point of commutative algebra, many additional aspects arise. In particular, there is a natural way to associa...

We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we a...

Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...