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ArticleOriginal scientific text

Title

Solutions of x³+y³+z³=nxyz

Authors 1

Affiliations

  1. Royal Institute of Technology, S-10044 Stockholm, Sweden

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

Bibliography

  1. E. S. Barnes, On the diophantine equation x²+y²+c=xyz, J. London Math. Soc. 28 (1953), 242-244.
  2. M. Craig, Integer values of Σ(x²/yz), J. Number Theory 10 (1978), 62-63.
  3. E. Dofs, On some classes of homogeneous ternary cubic diophantine equations, Ark. Mat. 13 (1975), 29-72.
  4. E. Dofs, On extensions of 1 chains, Acta Arith. 65 (1993), 249-258.
  5. W. H. Mills, A method for solving certain diophantine equations, Proc. Amer. Math. Soc. 5 (1954), 473-475.
  6. S. P. Mohanty, A system of cubic diophantine equations, J. Number Theory 9 (1977), 153-159.
  7. L. J. Mordell, Diophantine Equations, Academic Press, New York, 1969.
  8. E. G. Straus and J. D. Swift, The representation of integers by certain rational forms, Amer. J. Math. 78 (1956), 62-70.
  9. E. Thomas and A. T. Vasquez, Diophantine equations arising from cubic number fields, J. Number Theory 13 (1981), 398-414.
Acta Arithmetica
1995/73/3
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Pages:
201-213
Main language of publication
English
Received
1994-02-24
Accepted
1994-09-05
Published
1995
Exact and natural sciences