The integral points on elliptic curves $y^2=x^3+(36n^2-9)x-2(36n^2-5)$

Yang, Hai, and Fu, Ruiqin. "The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$." Czechoslovak Mathematical Journal 63.2 (2013): 375-383. <http://eudml.org/doc/260678>.

@article{Yang2013,
abstract = {Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$ has only the integral point $(x, y)=(2, 0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.},
author = {Yang, Hai, Fu, Ruiqin},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; integral point; quadratic diophantine equation; elliptic curve; integral point; cubic Diophantine equation},
language = {eng},
number = {2},
pages = {375-383},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$},
url = {http://eudml.org/doc/260678},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Yang, Hai
AU - Fu, Ruiqin
TI - The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 375
EP - 383
AB - Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$ has only the integral point $(x, y)=(2, 0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.
LA - eng
KW - elliptic curve; integral point; quadratic diophantine equation; elliptic curve; integral point; cubic Diophantine equation
UR - http://eudml.org/doc/260678
ER -