The Diophantine equations $ x^{n}_{1} +x^{n}_{2} +...+x^{n}_{r_{1}}= y ^{n}_{1} +y^{n}_{2} +...+y^{n}_{r_{2}} $
| dc.creator | Ivanov, Michael A. | |
| dc.date | 2004-12-01 | |
| dc.date.accessioned | 2026-07-07T05:14:53Z | |
| dc.date.available | 2026-07-07T05:14:53Z | |
| dc.description | 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. | |
| dc.description | 11 pages, no figure, Latex. Eprint of published paper of 1996 | |
| dc.identifier | https://arxiv.org/abs/math/0412036 | |
| dc.identifier | http://arxiv.org/abs/math/0412036 | |
| dc.identifier | Rend. Sem. Mat. Univ. Pol. Torino, Vol. 54, 1 (1996) pp 25-33 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73455 | |
| dc.subject | Number Theory | |
| dc.subject | Commutative Algebra | |
| dc.title | The Diophantine equations $ x^{n}_{1} +x^{n}_{2} +...+x^{n}_{r_{1}}= y ^{n}_{1} +y^{n}_{2} +...+y^{n}_{r_{2}} $ | |
| dc.type | text |