An elliptic curve having large integral points

Yanfeng He; Wenpeng Zhang

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 4, page 1101-1107
  • ISSN: 0011-4642

Abstract

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The main purpose of this paper is to prove that the elliptic curve E : y 2 = x 3 + 27 x - 62 has only the integral points ( x , y ) = ( 2 , 0 ) and ( 28844402 , ± 154914585540 ) , using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.

How to cite

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He, Yanfeng, and Zhang, Wenpeng. "An elliptic curve having large integral points." Czechoslovak Mathematical Journal 60.4 (2010): 1101-1107. <http://eudml.org/doc/196478>.

@article{He2010,
abstract = {The main purpose of this paper is to prove that the elliptic curve $E\colon y^2=x^3+27x-62$ has only the integral points $(x, y)=(2, 0)$ and $(28844402, \pm 154914585540)$, using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.},
author = {He, Yanfeng, Zhang, Wenpeng},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; integral point; Diophantine equation; elliptic curve; integral point; Diophantine equation},
language = {eng},
number = {4},
pages = {1101-1107},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An elliptic curve having large integral points},
url = {http://eudml.org/doc/196478},
volume = {60},
year = {2010},
}

TY - JOUR
AU - He, Yanfeng
AU - Zhang, Wenpeng
TI - An elliptic curve having large integral points
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 4
SP - 1101
EP - 1107
AB - The main purpose of this paper is to prove that the elliptic curve $E\colon y^2=x^3+27x-62$ has only the integral points $(x, y)=(2, 0)$ and $(28844402, \pm 154914585540)$, using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.
LA - eng
KW - elliptic curve; integral point; Diophantine equation; elliptic curve; integral point; Diophantine equation
UR - http://eudml.org/doc/196478
ER -

References

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  1. Baker, A., 10.1112/jlms/s1-43.1.1, J. Lond. Math. Soc. 43 (1968), 1-9. (1968) Zbl0157.09801MR0231783DOI10.1112/jlms/s1-43.1.1
  2. Stroeker, R. J., Tzanakis, N., 10.1080/10586458.1999.10504395, Exp. Math. 8 (1999), 135-149. (1999) Zbl0979.11060MR1700575DOI10.1080/10586458.1999.10504395
  3. Stroeker, R. J., Tzanakis, N., 10.1090/S0025-5718-03-01497-2, Math. Comput. 72 (2003), 1917-1933. (2003) Zbl1089.11019MR1986812DOI10.1090/S0025-5718-03-01497-2
  4. Zagier, D., 10.1090/S0025-5718-1987-0866125-3, Math. Comput. 48 (1987), 425-436. (1987) Zbl0611.10008MR0866125DOI10.1090/S0025-5718-1987-0866125-3
  5. Walker, D. T., 10.1080/00029890.1967.11999992, Am. Math. Mon. 74 (1967), 504-513. (1967) MR0211954DOI10.1080/00029890.1967.11999992
  6. Walsh, G., 10.1007/s000130050376, Arch. Math. 73 (1999), 119-125. (1999) MR1703679DOI10.1007/s000130050376

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