The integral points on elliptic curves

Hai Yang; Ruiqin Fu

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 375-383
  • ISSN: 0011-4642

Abstract

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Let be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if and both and are odd primes, then the general elliptic curve has only the integral point . By this result we can get that the above elliptic curve has only the trivial integral point for etc. Thus it can be seen that the elliptic curve really is an unusual elliptic curve which has large integral points.

How to cite

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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 -

References

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