Algorithm to solve certain ternary quartic Diophantine equations

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

In this paper, we develop an algorithm to solve completely the Diophantine equation F(x,y)=z2, where the quartic inhomogeneous polynomial F(x,y) with integer coefficients satisfies certain technical conditions. The procedure is an extension of the version of Runge's method given by Poulakis.

Anahtar Kelimeler:

Diophantine equation, Runge method, inhomogeneous ternary equation

Algorithm to solve certain ternary quartic Diophantine equations

Keywords:

Diophantine equation, Runge method, inhomogeneous ternary equation,

PDF

___

Beukers, F., Tengely, Sz.: An implementation of Runge’s method for Diophantine equations. arXiv:math/0512418v1. Hilliker, D. E., Strauss, E. G.: Determination of bounds for the solutions to those Diophantine equations that satisfy the hypotheses of Runges theorem. Trans. Amer. Math. Soc. 280, 637-657 (1983).
Mordell, L., J.: On some ternary quartic Diophantine equations. Elem. Math. 31, 89-90 (1966).
Poulakis, D.: A simple method for solving the Diophantine equation Y 2 = X 4 + aX 3 + bX 2 + cX + d . Elem. Math. 54, 32-36 (1999).
Runge, C.: ¨ Uber ganzzahlige L¨ osungen von Gleichungen zwissen zwei Ver¨ anderlichen. J. Reine Angew. Math. 100, 425-435 (1887).
Szalay, L.: Fast algorithm for solving superelliptic equations of certain types. Acta Acad. Paed. Agriensis 27, 19-24 (2000).
Szalay, L.: Superelliptic equations of the form y p = x kp + a kp −1 x kp −1 + · · · + a . Bull. Greek Math. Soc. 46, 23-3 (2002).
Tzanakis, N.: Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations. Acta Arithm. 75, 165-190 (1996).
Walsh, P. G.: A quantitative version of Runge’s theorem on Diophantine equations. Acta Arithm. 62, 157-172 (1992).