On Cornacchia’s algorithm for solving the diophantine equation

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

On the Diophantine Equation x^6+ky^3=z^6+kw^3

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...

An algorithm for solving a diophantine equation with lower and upper bounds on the variables

We develop an algorithm for solving a diophantine equation with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction, and rst nds short vectors satisfying the diophantine equation. The next step is to branch on linear combinations of these vectors, which either yields a vector that satis es the bound constraints or provides a proof that no such vector exist...

On Solving Linear Diophantine Systems Using Generalized Rosser’s Algorithm

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...

Short solutions for a linear Diophantine equation

It is known that finding the shortest solution for a linear Diophantine equation is a NP problem. In this paper we have devised a method, based upon the basis reduction algorithm, to obtain short solutions to a linear Diophantine equation. By this method we can obtain a short solution (not necessarily the shortest one) in a polynomial time. Numerical experiments show superiority to other method...