Solutions of x³+y³+z³=nxyz

Erik Dofs

Acta Arithmetica (1995)

  • Volume: 73, Issue: 3, page 201-213
  • ISSN: 0065-1036

Abstract

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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

How to cite

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Erik Dofs. "Solutions of x³+y³+z³=nxyz." Acta Arithmetica 73.3 (1995): 201-213. <http://eudml.org/doc/206818>.

@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 -

References

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  1. [1] E. S. Barnes, On the diophantine equation x²+y²+c=xyz, J. London Math. Soc. 28 (1953), 242-244. Zbl0050.03701
  2. [2] M. Craig, Integer values of Σ(x²/yz), J. Number Theory 10 (1978), 62-63. 
  3. [3] E. Dofs, On some classes of homogeneous ternary cubic diophantine equations, Ark. Mat. 13 (1975), 29-72. Zbl0301.10021
  4. [4] E. Dofs, On extensions of 1 chains, Acta Arith. 65 (1993), 249-258. 
  5. [5] W. H. Mills, A method for solving certain diophantine equations, Proc. Amer. Math. Soc. 5 (1954), 473-475. Zbl0055.27104
  6. [6] S. P. Mohanty, A system of cubic diophantine equations, J. Number Theory 9 (1977), 153-159. Zbl0349.10010
  7. [7] L. J. Mordell, Diophantine Equations, Academic Press, New York, 1969. 
  8. [8] E. G. Straus and J. D. Swift, The representation of integers by certain rational forms, Amer. J. Math. 78 (1956), 62-70. Zbl0070.04402
  9. [9] E. Thomas and A. T. Vasquez, Diophantine equations arising from cubic number fields, J. Number Theory 13 (1981), 398-414. Zbl0468.10009

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