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

Hai Yang; Ruiqin Fu

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 3, page 853-862
  • ISSN: 0011-4642

Abstract

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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 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p ≥ 17 , then N ( p ) ≤ 4 or 1 depending on whether p ≡ 1 ( mod 8 ) or p ≡ - 1 ( mod 8 ) .

How to cite

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Yang, Hai, and Fu, Ruiqin. "Integral points on the elliptic curve $y^2=x^3-4p^2x$." Czechoslovak Mathematical Journal 69.3 (2019): 853-862. <http://eudml.org/doc/294510>.

@article{Yang2019,
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 $E\colon y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N(p)\le 4$ or $1$ depending on whether $p\equiv 1\hspace\{4.44443pt\}(\@mod \; 8)$ or $p\equiv -1\hspace\{4.44443pt\}(\@mod \; 8)$.},
author = {Yang, Hai, Fu, Ruiqin},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; integral point; quadratic equation; quartic Diophantine equation},
language = {eng},
number = {3},
pages = {853-862},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integral points on the elliptic curve $y^2=x^3-4p^2x$},
url = {http://eudml.org/doc/294510},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Yang, Hai
AU - Fu, Ruiqin
TI - Integral points on the elliptic curve $y^2=x^3-4p^2x$
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 3
SP - 853
EP - 862
AB - 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 $E\colon y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N(p)\le 4$ or $1$ depending on whether $p\equiv 1\hspace{4.44443pt}(\@mod \; 8)$ or $p\equiv -1\hspace{4.44443pt}(\@mod \; 8)$.
LA - eng
KW - elliptic curve; integral point; quadratic equation; quartic Diophantine equation
UR - http://eudml.org/doc/294510
ER -

References

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