Diophantine equations

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 + b , a − b , 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) = r1 + r2 − 1 for solvable equations of such a type as r(n) 6 2n + 1 (recently the better one with r(n) 6 2n − 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 6 11 which were found with the help of the algorithm that is not connected with linearization.