Advance of algebraic approach to solving systems of Diophantine equations
by
Yuri Chebrakov
Department of Mathematics, Technical University, St.-Petersburg, Russia

It is a well-known fact that a general algebraic solution of linear Diophantine equations systems may be found by standard algebraic methods (for instance, by the Gauss method). If the algebraic solution contains a large number of arbitrary selected parameters, it is useless for obtaining concrete solutions of a given set of Diophantine equations (the limit of the quantity of arbitrary selected parameters is determined by the requirement to perform calculations in some finite time). We demonstrate in this talk that a translation of the description of the general algebraic solution into the language of numerical sequences is sometimes sufficient for overcoming this computative difficulty. If this way is insufficient, we suggest to replace the general algebraic solution containing L arbitrary selected parameters by a set of algebraic solutions containing less than L parameters. In such a method, the main difficulty consists in proving the completeness of the set of algebraic solutions. By use of both methods, we find some new solutions for a few combinatorial problems on Magic Squares.

Date received: March 15, 1999