## Ajai Choudhry - Some diophantine problems concerning equal sums of integers and their cubes

hrj:171 - Hardy-Ramanujan Journal, January 1, 2010, Volume 33 - 2010 - https://doi.org/10.46298/hrj.2010.171
Some diophantine problems concerning equal sums of integers and their cubes

Authors: Ajai Choudhry

This paper gives a complete four-parameter solution of the simultaneous diophantine equations $x+y+z=u+v+w, x^3+y^3+z^3=u^3+v^3+w^3,$ in terms of quadratic polynomials in which each parameter occurs only in the first degree. This solution is much simpler than the complete solutions of these equations published earlier. This simple solution is used to obtain solutions of several related diophantine problems. For instance, the paper gives a parametric solution of the arbitrarily long simultaneous diophantine chains of the type $x^k_1+y^k_1+z^k_1=x^k_2+y^k_2+z^k_2=\ldots=x^k_n+y^k_n+z^k_n=\ldots,~~k=1,3.$ Further, the complete ideal symmetric solution of the Tarry-Escott problem of degree $4$ is obtained, and it is also shown that any arbitrarily given integer can be expressed as the sum of four distinct nonzero integers such that the sum of the cubes of these four integers is equal to the cube of the given integer.

Volume: Volume 33 - 2010
Published on: January 1, 2010
Imported on: March 3, 2015
Keywords: equal sums of powers,equal sums of cubes,Tarry-Escott problem,diophantine chains Mathematics,[MATH] Mathematics [math]