# Article

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Keywords:
elliptic curve; integral point; Diophantine equation
Summary:
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.
References:
[1] Baker, A.: The Diophantine equation $y^2=ax^3+bx^2+cx+d$. J. Lond. Math. Soc. 43 (1968), 1-9. DOI 10.1112/jlms/s1-43.1.1 | MR 0231783 | Zbl 0157.09801
[2] Stroeker, R. J., Tzanakis, N.: On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement. Exp. Math. 8 (1999), 135-149. DOI 10.1080/10586458.1999.10504395 | MR 1700575 | Zbl 0979.11060
[3] Stroeker, R. J., Tzanakis, N.: Computing all integer solutions of a genus $1$ equation. Math. Comput. 72 (2003), 1917-1933. DOI 10.1090/S0025-5718-03-01497-2 | MR 1986812 | Zbl 1089.11019
[4] Zagier, D.: Large integral points on elliptic curves. Math. Comput. 48 (1987), 425-436. DOI 10.1090/S0025-5718-1987-0866125-3 | MR 0866125 | Zbl 0611.10008
[5] Walker, D. T.: On the Diophantine equation $mx^2-ny^2=\pm 1$. Am. Math. Mon. 74 (1967), 504-513. DOI 10.1080/00029890.1967.11999992 | MR 0211954
[6] Walsh, G.: A note on a theorem of Ljunggren and the Diophantine equations $x^2-kxy^2+y^4=1,4$. Arch. Math. 73 (1999), 119-125. DOI 10.1007/s000130050376 | MR 1703679

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