@article{ErikDofs1995,
abstract = {
The diophantine equation
(1) x³ + y³ + z³ = nxyz
has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive "strings" ([9]) or as (a few) parametric solutions ([3], [9]).
For a fixed n-value, (1) can be transformed into an elliptic curve with a recursive solution structure derived by the "chord and tangent process". Here we treat (1) as a quaternary equation and give new methods to generate infinite chains of solutions from a given solution \{x,y,z,n\} by recursion. The result of a systematic search for parametric solutions suggests a recursive structure in the general case.
If x, y, z satisfy various divisibility conditions that arise naturally, the equation is completely solved in several cases
},
author = {Erik Dofs},
journal = {Acta Arithmetica},
keywords = {cubic diophantine equations; quadratic diophantine equations; numerical solutions; parametric solutions; table},
language = {eng},
number = {3},
pages = {201-213},
title = {Solutions of x³+y³+z³=nxyz},
url = {http://eudml.org/doc/206818},
volume = {73},
year = {1995},
}
TY - JOUR
AU - Erik Dofs
TI - Solutions of x³+y³+z³=nxyz
JO - Acta Arithmetica
PY - 1995
VL - 73
IS - 3
SP - 201
EP - 213
AB -
The diophantine equation
(1) x³ + y³ + z³ = nxyz
has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive "strings" ([9]) or as (a few) parametric solutions ([3], [9]).
For a fixed n-value, (1) can be transformed into an elliptic curve with a recursive solution structure derived by the "chord and tangent process". Here we treat (1) as a quaternary equation and give new methods to generate infinite chains of solutions from a given solution {x,y,z,n} by recursion. The result of a systematic search for parametric solutions suggests a recursive structure in the general case.
If x, y, z satisfy various divisibility conditions that arise naturally, the equation is completely solved in several cases
LA - eng
KW - cubic diophantine equations; quadratic diophantine equations; numerical solutions; parametric solutions; table
UR - http://eudml.org/doc/206818
ER -
