Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 853-862, 2019

Integral points on the elliptic curve $y^2=x^3-4p^2x$

Hai Yang, Ruiqin Fu

Received November 21, 2017. Published online March 26, 2019.

Abstract: Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\geq17$, then $N(p)\leq4$ or $1$ depending on whether $p\equiv1\pmod8$ or $p\equiv-1\pmod8$.

Keywords: elliptic curve; integral point; quadratic equation; quartic Diophantine equation

Classification MSC: 11G05, 11D25, 11Y50

DOI: 10.21136/CMJ.2019.0529-17

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Hai Yang, School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, P. R. China, e-mail: xpuyhai@163.com; Ruiqin Fu, School of Science, Xi'an Shiyou University, Xi'an, Shaanxi, 710065, P. R. China, e-mail: xsyfrq@163.com

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