On a linear Diophantine equation and its applications

The paper considers a linear Diophantine equation with six variables. Its solution is constructed as a shifted incomplete five-dimensional integer lattice in a six-dimensional space.
The basis of this lattice is constructed.
An algorithmic solution for finding all its solutions from a given six-dimensional integer parallelepiped is given. For this purpose, a new basis for this incomplete five-dimensional lattice was constructed, which allowed us to write an effective program for finding all sets that satisfy a given Diophantine equation and belong to a given rectangular parallelepiped.
As a result of the proposed algorithm implemented in the Mathcad system, it was shown that out of the total number of 10182290760 integer points lying in a given parallelepiped, only
7822045 satisfy the given Diophantine equation. Thus, the total search was reduced by 1301.7 times.
The article considers the relationship between shifted lattices and integer programming problems. It is shown how it is possible to construct bases of incomplete integer lattices, which make it possible to reduce a complete search over the points of an s-dimensional rectangular parallelepiped to a search over the points of a shifted incomplete lattice lying in this parallelepiped.
Some applications of this Diophantine equation in technical issues related to the solution of an applied mechanical engineering problem in the field of measuring tool design, in particular, sets of end length measures, are considered.
The article reflects the iterative nature of the refinement of the mathematical model of this applied problem. After the first model adjustment, the number of sets decreased by another
193.237 times, and after the second model adjustment, the total reduction of sets suitable for subsequent optimization became 581114.6 times.
In conclusion, the directions of further research and possible application of the ideas of Hopfield neural networks and machine learning for the implementation of the selection of optimal
solutions are indicated.

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