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

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

2004-12-01

arxiv.org/abs/math/0412036v1

Rend. Sem. Mat. Univ. Pol. Torino, Vol. 54, 1 (1996) pp 25-33

Mathematics

Number Theory

11 pages, no figure, Latex. Eprint of published paper of 1996

Scientific paper

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.

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