The Diophantine equations $ x^{n}_{1} +x^{n}_{2} +...+x^{n}_{r_{1}}= y ^{n}_{1} +y^{n}_{2} +...+y^{n}_{r_{2}} $

Abstract

The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For $n=2$ such a procedure is well known, when new variables are components of spinors and they are widely used in mathematical physics. For example, parametrization of Pythagoras threes $a^{2} +b^{2}$, $a^{2} -b^{2}$, $2ab$ may be cited as an example in number theory where two independent variables form a spinor which can be obtained by solution of a system of two linear equations. We also investigate the combinatorial estimate for the smallest sum $r(n)=r _{1}+r_{2} -1 $ for solvable equations of such a type as $r(n) \leq 2n+1$ (recently the better one with $r(n) \leq2n-1$ was received by L. Habsieger (J. of Number Theory 45 (1993) 92)). Apart from that we consider two conjectures about $r(n)$ and particular solutions for $n \leq11$ which were found with the help of the algorithm that is not connected with linearization.