Showing the non-existence of solutions in systems of linear Diophantine equations

On the existence of solutions in systems of linear Diophantine equations

“…There are as well algorithms for computing the Smith normal form, both probabilistic [8] and deterministic [16], some of which are designed to be implemented on paralell machines [12]. In [9], we proposed a different method (which was not based in the calculation of the Smith formal form of a matrix) for detecting quickly the non-existence of solutions for a system of Diophantine linear equations through the calculation of certain functions named 'Testers'. Testers are linear functions defined on either the ring Z or the ring Z m providing necessary conditions so that a system of Diophantine equations has solution.…”

“…Testers are linear functions defined on either the ring Z or the ring Z m providing necessary conditions so that a system of Diophantine equations has solution. In [9], we also studied a method for obtaining easily testers defined on Z and the fields Z p , where p is a prime integer. However, questions about how to calculate testers defined on Z m , where m is not prime, was not then considered.…”

“…However, questions about how to calculate testers defined on Z m , where m is not prime, was not then considered. Besides, in [9] we made clear that testers may be useful for proving the non-existence of solutions in Diophantine linear systems, but we did not use them to prove the existence of solutions.…”

“…The work[9] presents a method for finding those prime integers p which provide useful testers in Z p . In this way, in[9] we provide a way for finding all the testers defined in Z and Z p , where p is any prime integer.…”

New methods for proving the impossibility to solve problems through reduction of problem spaces

“…Of course, the main concern about applying this technique is related to the need of finding suitable reductions, which sometimes may turn out to be a difficult task. In this paper we provide a general method, based on previous work on Diophantine equations [14] for finding such suitable reductions in a special kind of problem spaces: namely, linear problem spaces.…”