The use of Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms for obtaining proofs of unsatisfiability and interpolants for conjunctions of line...

A Piatetski-Shapiro sequence with exponent $\alpha$ is a of integer parts $n^\alpha$ $(n = 1,2,\ldots)$ non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set those terms. In this article, we study so that equation $ax + by cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give lower bound for its Hausdorff dimension. As corollary, find...

An arbitrary quadratic diophantine equation with two unknowns can be reduced to a Pell-type equation. How can such equations be solved? Recall that the general solution of a linear diophantine equation is a linear function of some parameters. This does not happen with general quadratic diophantine equations. However, as we will see later, in the case of such equations with two unknowns there st...

In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Con-tejean and Devie 9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm o...