IJPAM: Volume 21, No. 3 (2005)

ON THE DIOPHANTINE EQUATION $X^4- DY^4=\pm Z^2$
AND ITS ASSOCIATED ELLIPTIC CURVE $V^2=U^3-DU$

Omar Kihel$^1$, Claude Levesque$^2$
$^1$Department of Mathematics
Brock University
St. Catharines, Ontario, L2S 3A1, CANADA
e-mail: okihel@brocku.ca
$^2$Département de Mathématiques et de Statistique
Centre Interuniversitaire en Calcul Mathématique Algébrique (CICMA)
Université Laval
Québec, G1K 7P4, CANADA
e-mail: cl@mat.ulaval.ca


Abstract.We plan to study the Diophantine equations $\,X^4 - DY^4 = Z^2 \; \mbox{and} \; x^4 - Dy^4 = -z^2,\,$ where $\,D \in {\bf Z}.\, $ In particular, when $\,D=p\,$ or $\,-p,\,$ where $p$ is a prime, we give a characterization of all the solutions of $\,X^4 - DY^4 = Z^2.\,$ We conclude with exploiting some links between the title equation and the associated elliptic curve $\,E: V^2=U^3-DU.\,$

Received: May 5, 2005

AMS Subject Classification: 11D25, 11G05

Key Words and Phrases: Diophantine equations, quartic equations, elliptic curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 3