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On extensions of 1 chains

100%

Dofs E.

Acta Arithmetica

1993

tom 65

nr 3

249-258

Solutions of x³+y³+z³=nxyz

100%

Dofs E.

Acta Arithmetica

1995

tom 73

nr 3

201-213

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

Ograniczanie wyników

2 Acta Arithmetica

2 Dofs E.

1 1995

1 1993

Wyniki wyszukiwania

On extensions of 1 chains

Solutions of x³+y³+z³=nxyz